Diversification Benefits of Catastrophe Bonds:

Master’s Thesis EORAS Specialization: Actuarial Science

Diversification Benefits of Catastrophe Bonds:

an Empirical Study using a Copula and Copula- GARCH framework January 30, 2018

Author:

G Haver

Supervisors:

Prof. Dr. R. H. Koning (University of Groningen) V. Stap (EY Risk Department) Co-Assessor: Prof. Dr. T. K. Dijkstra (University of Groningen) Abstract

Acknowledgement

With this thesis I would like to conclude my masters degree in Econometrics, Operations Research and Actuarial Studies, with a specialization is Actuarial studies, at the University of Groningen.

First and foremost I would like to thank my supervisor Professor Ruud Koning. His guidance and support were of great value, as he let me perform my own research, but steered me in the right direction when needed. Furthermore, I would like to express my gratitude to Vincent Stap from the Risk department of EY. His personal experiences with catastrophe bonds were very helpful. Also I would like to thank Eveline Takken-Somers and Wouter Aarts from PGGM, for taking the time to show me how catastrophe bonds are used in practice. Pleasant to see two people talk so passionately about their line of work. Lastly, I would like to thank my parents for their unconditional support during my studies.

Contents

1 Introduction 4

2 Structure of Catastrophe Bonds 7

3 Data Analysis 7 4 Methodology 12 5 Empirical Analysis 19 6 Conclusion 31 7 Discussion 32 8 References 34 9 Appendices 36 9.1 Appendix A . . . 36 9.2 Appendix B . . . 45

9.2.1 Mean Excess function . . . 45

1

Introduction

In August 2005 Katrina, a category 5 Hurricane, hit the Southern states of the United States of America. After raging through the Caribbean islands it destroyed huge parts of Texas, Louisiana and Arkansas, leaving countless people homeless and taking 1836 lives. Mayor cities like Houston, Jackson and New Orleans were ruined. Events like these hugely influence the insurance and re-insurance markets. As hurricanes are not an uncommon phenomenon in the Southern parts of the US, many people are insured against these catastrophes. A natural disaster hence leads to billions of dollars in insured losses. Most insurance compa-nies do not have the provisions to account for this and would hence go bankrupt. Luckily they have thought of a solution, in the shape of catastrophe (CAT) bonds. In short these bonds work as follows; investors can buy these bonds, describing a specific catastrophe in a predetermined location. The insurance company will use this investment as collateral for the disaster. When this catastrophe does not occur within the maturity of the bond, investors will receive returns from the collateral account, including both its principal and a premium paid from the insurance company for providing coverage. Whenever the catastrophe does occur however, investors lose their principal and the money will be used to cover the damage. More information about the structure of these bonds will be provided in Section 2.

Most of the current literature describes methodologies used to price CAT bonds. These securities are incredibly hard to value, as they are highly dependent on time and location. Data collection would be very extensive and beyond the scope of this paper. Instead of investigating the pricing structure we will obtain a CAT bond index, and assume that the index is priced correctly. Subsequently, we examine influence of this bond within a investor’s portfolio. CAT bonds are initially designed to only capture event risk. This would imply no external financial influences. These so called zero-beta investments are known to have high diversification benefits, which brings us to the research question considered in this paper. Does the presence of catastrophe bonds within a portfolio bring diversification benefits? To answer this question, two subquestions are constructed.

1. Are the CAT bonds zero-beta investments, such as initially intended?

2. Do the diversification benefits differ in crisis periods related to non-crisis periods?

(VaR) of CAT bonds is examined, conditional that the other assets incurred a loss equal to their 95% VaR. This metric, proposed by Kole, Koedijk, and Verbeek (2007), will henceforth be called “conditional VaR”. Plotting this VaR against confidence interval α ∈ (0, 1) will provide insight in the behavior of CAT bonds given that the other assets incur a high loss. Furthermore, it is of importance to investigate whether these potential benefits are present during crisis periods, when they are needed most. To answer the second subquestion we have to investigate the dynamics of the two metrics. Therefore, three periods will be constructed: a pre-crisis, crisis, and post-crisis period. The behavior of the copula parameters and condi-tional VaR throughout the crisis will provide insight the influence of the economic crisis on the diversification benefits.

A drawback of this approach is that the results are dependent on the specification of a crisis period. Moreover, correlation trends within periods can not be observed. Therefore, the analysis is extended using a time-varying t copula which uses a DCC-GARCH structure in its correlation matrix. This way correlations can be observed throughout the crisis without specifying periods.

literature review

Literature is scarcely available that focuses on the diversification benefits of CAT bonds. One of the papers that does focus on the diversification benefits of CAT bonds is Cum-mins and Weiss (2009). They use a time-invariant correlation matrix to investigate the bond returns throughout the financial crisis, and find that CAT bonds are close to zero-beta in-vestments under normal economic conditions. When in crisis however, they find that CAT bonds are significantly correlated to the other assets. Carayannopoulos and Perez (2015) extend the analysis by removing the assumption of constant correlation. They use a multi-variate GARCH model to capture the time-varying volatility and use dynamic hedge ratios to investigate correlation throughout the financial crisis. If CAT bonds have very low in-vestment betas then they can be used for diversification purposes1. Their results are in line with those of Cummins and Weiss (2009), showing no correlation in non-crisis periods, but not immune to the consequences of the financial crisis in 2008. They conclude that this vul-nerability to the market was a result of counterparty risk. However, the betas are relatively small in comparison with other assets such as equity and corporate bonds, implying stronger

1_{Beta is a measure used to determine correlation of an asset with to the total market. It can be defined as}

Beta = ρ(ra,rm)

σi

diversification benefits.

Both papers use linear correlation as a measure of dependence. As discussed in Section 4 this has several downsides, as this approach is only applicable when the underlying distribu-tion is elliptical. Instead we use a copula to measure dependence. The assets shall be modeled individually and joined using a t copula. This methodology is extensively discussed in Kole et al. (2007). They perform a copula investigation on a portfolio consisting three assets: real estate, equity and fixed income. Clear evidence is provided in favor of the Student’s t copula in comparison with the Gaussian and extreme value based Gumbel copula. Moreover, they propose to investigate diversification benefits using two metrics: The level of dependence, estimated using rank correlations, and the conditional VaR. These measures shall also be used to investigate the diversification benefits of CAT bonds. The dynamics are examined by comparing the metrics in three periods.

The results will then however rely on the specification of a crisis period, weakening the strength of our conclusions. Hence the analysis will be extended using a dynamic t copula discussed by Kang (2015). They use a copula-GARCH framework to model 45 US stock re-turns and show that a time-varying t copula has a significantly better fit than normal copula models. Furthermore, their model can capture time-varying conditional correlation, provid-ing insight in the dynamics of the diversification benefits. (Patton, 2006a) investigates the conditional dependence of exchange rates using a Copula-GARCH model, and shows that correlation increases in times of depression. This paper will investigate whether the same holds for CAT bonds, as their structure should isolate them from market risk.

2

Structure of Catastrophe Bonds

Before starting with the analysis it is important to understand the structure of CAT bonds. To overcome the need of keeping extreme provisions, insurance companies issue CAT bonds, bonds constructed in such a manner that the risk of a specified natural disaster is transferred from insurance company to investor. This structure is described using Figure 1.

Proceeds + Premiums Return on Collateral Liquidation of Assets Premiums

Reimbursement Cash Proceeds + Interest

Principal

Insurer/ Sponsor Special Purpose Vehicle CAT Bond Investor

Collateral Account

Figure 1: Structure of a CAT Bond. The dashed lines indicate flows are event contingent. This figure is based on a figure of Artemis (2017)

A Special Purpose Vehicle (SPV) is used as intermediary between the insurer/re-insurer (sponsor) and the investor. The SPV receives premiums from the sponsor for the service of providing coverage. Moreover, the SPV receives proceeds from the investors in exchange for the issued securities. The proceeds and premiums are deposited into a collateral account, where it is invested in high quality securities. To eliminate interest rate risk, and the effect of asset fluctuations, a total return swap is used such that the investors are only exposed to the risk of the underlying peril (Carayannopoulos and Perez (2015)). Contingent on a qualifying event the SPV can payout the sponsor by liquidating the assets. This reimbursement will only take place if the terms of the CAT bond transaction are met. When no event occurs within the maturity of the bonds, then the collateral is liquidated and investors receive the returns, consisting their principal, the premium paid by the insurer and interest on both (Artemis (2017)). Using this structure the CAT bond investors should be immune to market risk, resulting in diversification benefits. This statement will be verified in this paper. However, let us first analyze the data.

3

Data Analysis

and Real Estate (RE). For a complete analysis of diversification benefits a global benchmark variable is obtained for each type. Hence, a portfolio consisting of four different assets is constructed. CAT bond price indices are generally provided on a weekly basis. Following Cummins and Weiss (2009) we should focus on the broadest indices2. Carayannopoulos and Perez (2015) suggest to use the BB-rated index. This benchmark covers 47 of the biggest publicly traded CAT bonds, in total a market of 8bn USD (Swiss Re (2017)). More infor-mation about the composition of this index is provided in Figure 10 in Appendix A. Table 8 in Appendix A provides a description of each variable. As described the variables chosen in our portfolio are all in the broadest sense; global benchmark variables including both de-veloped and emerging markets. All data is extracted from Bloomberg, a reliable source. All variables are issued in USD and where needed hedged against currency risk. As global bench-mark variables are used, our conclusions are applicable for a large group of investors. Weekly data is obtained ranging from January 2002 to September 2017, resulting in 921 observations.

The black line in Figure 2 presents the development of the CAT bond index. As can be seen the index increases gradually, with irregular permanent decreases. To explain these shifts, all mayor catastrophes from 2002 to 2017 are displayed in Table 9 in Appendix A. Combining CAT bonds in Figure 2 with the observations in Table 9 we can conclude that the decreases are related to a mayor catastrophe. Note that these drops are caused by in-sured losses, not deaths. As can be seen, the Indonesian tsunami in December 2004, with more than 200,000 victims, did not have any impact on the index, while Katrina and the Japanese tsunami, with insured losses of 82.31 and 38.09bn respectively, had huge influence. A remarkable decrease is the one in September 2008. This could be caused by hurricane Ike, which resulted in 23.03bn insured losses in the US. However, the collapse of Lehman Brothers occurred simultaneously. The cause of this drop is therefore debatable, and will be discussed in Section 5.

Also the other asset indices are presented in Figure 2. For a proper comparison, all assets are normalized to index 100 with reference date the fourth of January 2002. As can be seen the RE and equity markets are heavily influenced by the financial crisis, as evident decreases are depicted in 2008. Considering the two bonds, a different trend is observed. Even though drops are perceived in April and September for government and CAT bonds respectively, they are

2

Figure 2: Index development

not as vigorous as those observed in the equity and RE market. Besides several contractions a rather constant positive trend is observed. This confirms our hypothesis, stating limited dependence with the other assets. The slope, especially after the financial crisis, shows to be more moderate for government bonds than for CAT bonds, implying smaller payoff. For analytical purposes it is more convenient to investigate returns instead of the price level. Therefore, the data is transformed to log-differences. The summary statistics are presented in Table 1. These stylized facts are used to construct our marginal distributions.

Table 1: Descriptive Statistics of the weekly returns (%)

Descriptive stats CAT FTSE T-Bond Real Estate

Min −4.086 −22.431 −1.443 −18.466 Mean 0.108 0.076 0.072 0.078 Max 4.292 11.529 1.561 17.237 Std. dev. 0.364 2.254 0.369 2.697 Skewness −2.645 −1.312 −0.309 −0.869 Kurtosis 61.469 16.160 3.841 13.694

occurrence of a catastrophe. As variance relies on the sum of deviations, this results in a similar variance measure, but a different behavior. At these negative peaks, which are also represented by the lower minimum value in Table 1, the investor risks losing his principal. To compensate, a premium is paid by the insurance companies. The collateral account consists both the principal and this premium. Contingent on no trigger event the principal, premium and interest will be returned to the investor3. As this premium is generally higher than the long-run expected loss due to catastrophes, a higher mean return is observed for CAT bonds.

CAT returns Frequency −4 −2 0 2 4 0 100 200 300 FTSE returns Frequency −4 −2 0 2 4 0 40 80 120 T−Bond returns Frequency −4 −2 0 2 4 0 20 40 60

Real Estate returns

Frequency −4 −2 0 2 4 0 40 80 120

Figure 3: Histograms of the four assets

The histograms in Figure 3 clearly represent the variance presented in Table 1. Both bonds show a smaller spread than the FI asset (FTSE), and RE asset (REIT). To have a consistent scaling on the x-axis the tails of the FTSE and REIT are not depicted. The full histograms are therefore presented in Figure 12 in Appendix A. Comparison of these four his-tograms with Table 1 yields some interesting insights. First of all the high skewness presented in Table 1 is not depicted. As can be seen the CAT and FTSE show to be rather symmetrical, while the statistics show a high negative skewness. We might even argue that the CAT bond histogram is positively skewed. Reasoning for the high negative skewness can be found in the extreme events. As Figure 2 shows, CAT bonds indices incur high permanent decreases when a disaster occurs, but rarely shows exceptionally high increases. These events heavily influ-ence skewness. Excluding 2.5% extremes on both sides shows that skewness of CAT bonds changes from -2.65 to 0.13. We can conclude that the underlying distribution is not high negatively skewed and that this skewness is due to extreme events. To a lesser extend this is also the case for FTSE, where symmetry is observed in the histogram and skewness is related

3

to extreme events in the left tail. From this we can conclude that despite the high negative skewness in Table 1, symmetric marginals might fit better for CAT bonds and FTSE. The T-Bonds and RE show skewed histograms, implying that implementing asymmetry might enhance fit when modeling these marginals. Also the high kurtosis can be described. As can be seen in Figure 11, an enlarged version of the CAT bond histogram, there is mass in the tails of the distribution, while the variance is low. High deviations from the mean in combination with a low variance result in a very high kurtosis. Also the FTSE and REIT show more mass in the tails, implying that marginal models should be selected that are able to describe fat tails. To formally reject normality Jarque-Bera tests are presented in Table 10 in Appendix A. As the p-value of every test is very close to zero we reject normality.

−20 −10 0 10 2003 2005 2007 2009 2011 2013 2015 2017 Date CA T −20 −10 0 10 2003 2005 2007 2009 2011 2013 2015 2017 Date T−Bond −20 −10 0 10 2003 2005 2007 2009 2011 2013 2015 2017 Date FTSE −20 −10 0 10 2003 2005 2007 2009 2011 2013 2015 2017 Date Real Estate

Figure 4: Returns over time of all four assets

Figure 4 shows the asset dynamics. As can be seen the volatility of CAT bonds, FTSE and REIT shows to be unstable, as variance is not constant over time. When constructing dynamic marginals, this suggests to use a GARCH-part to describe the time-varying volatil-ity. The Ljung-Box tests in Table 11 support this, as despite for the government bond, all volatilities show to be serially correlated. Figure 14 in Appendix A depicts the plots of the autocorrelations and partial autocorrelations respectively. As can be seen, only the plots corresponding to the CAT bonds show to tail off gradually. This implies that the conditional mean of CAT bonds is properly described using lagged values, in contrast with the other assets. Also these observations are used when constructing dynamic marginals.

4

Methodology

Two metrics are used to investigate the diversification benefits. First, the copula parameters that represent dependence between CAT bonds and other assets are examined. Second, a VaR conditional on the other losses incurring a high loss is plotted. The level of these metrics will be used to answer our first subquestion, examining whether CAT bonds behave indepen-dently of other assets. Plotting the conditional VaR provides a visual representation of these diversification benefits in times of extreme losses. The dynamics of these metrics will provide insight in answering our second subquestion, investigating whether the diversification benefits differ in crisis related to non-crisis periods. To obtain these measures and to investigate their dynamics, the methodologies of Cummins and Weiss (2009), Kole et al. (2007) and Kang (2015) will be combined.

Pearson Correlation

To get a first sight of our problem the results of Cummins and Weiss (2009) will be replicated. Their analysis, based on the Pearson correlation, is a simple but effective way to investigate linear relationships between two assets. To analyze the dynamics they define three periods: a pre-crisis, crisis and post-crisis period. For each period the Pearson correlation coefficients are estimated. Comparison provides insight in the sensitivity of CAT bonds to market changes. Furthermore, comparison of our results with those obtained by Cummins and Weiss (2009), shows validity of our dataset. If their results can be replicated using our set, then any de-viations later on can be appointed to methodological differences, and is not the result of a different set of variables.

The level of correlation will provide an answer to our first subquestion. To test whether CAT bonds are zero-beta investments we should test H0: ρi,(x,y)= 0. The following statistic

will be used: t-valuei,(x,y) = ρi,(x,y) q 1 − (ρi,(x,y))2 √ ni− 2, (1)

where ρi,(x,y)represents Pearson’s correlation coefficient between assets x and y, and ni is the

sample size in period i = 0, 1, 2, 3. Period 0 represents the total dataset, where the periods 1,2 and 3 represent the pre-crisis, crisis, and post-crisis periods respectively. This statistic will be tested against the t-table with dfi= ni− 2. The p-values indicate whether CAT bonds

For our second subquestion we need to examine whether the coefficients differ significantly between the period subsets. Therefore Fisher’s transformation is applied to the observations in i = 1, 2, 3. As the period subsets are independent, this approach is applicable. Define

z_i,(x,y) as the Fisher’s transformation:

z_i,(x,y) = 0.5 ln 1 + ρi,(x,y)

1 − ρ_i,(x,y). (2)

This new random variable is normally distributed. Define period j = 1, 2, 3. The hypothesis

H0: ρi,(x,y)− ρj,(x,y)= 0 where i 6= j can be tested using the following statistic:

t_i,j,(x,y)= (z_i,(x,y)− z_j,(x,y)) s

1

(ni− 3) + (nj− 3)

. (3)

On this statistic a z-test will be performed. The resulting p-value will show whether the correlations changed significantly between the pre-crisis, crisis and post-crisis period.

Using linear correlation as a measure of dependence has several downsides. When inves-tigating diversification benefits, the level of correlation might not fully explain the level of dependence, as zero correlation does not always imply independence between the assets. Fur-thermore, correlations are not invariant under nonlinear strictly increasing transformations of risk and are only well defined under finite variances (Embrechts, McNeil, and R¨udiger (2015)). These shortcomings imply that linear correlations only describe dependence well when the distribution belongs to an elliptical class, such as a Multivariate Normal.4 For financial instruments occasional extreme losses might skew the underlying distribution, mak-ing it debatable whether the underlymak-ing distribution is indeed elliptical. Therefore we apply a copula approach, joining marginals distributions such that the dependence is exhibited solely by the copula. Using this approach, dependence is well described for both ellipti-cal and non-elliptiellipti-cal distributions. Furthermore, the copula has the advantage that assets can be modeled individually, resulting in more flexible modeling techniques and a smaller misspecification error.

Static t Copula

The level of dependence will be obtained by a static t copula, which uses rank correlation. To verify whether the copula indeed represents the correct rank correlations the results will be compared to the Pearson’s correlation of the probability transformed data. The level of

4

dependence indicate whether there is presence of diversification benefits. Moreover, the VaR of CAT bonds will be obtained under the assumption that all other assets are in their 95% VaR. Consider the assets A and B. Then the VaR of asset A with confidence level α and loss L, given that B is in its 95% VaR is:

VaRα = inf{l ∈ R : FA(l) ≥ α|LB ≥ V aRB,95%} (4)

For our analysis, not only asset B is considered to be in its 95% VaR, but the FTSE, T-Bond and REIT jointly. Plotting this VaR for α ∈ (0, 1) provides insight in the diversification benefits of CAT bonds, given extreme losses of other assets. Both metrics provide insight in answering our first subquestion. For the second question the dynamics of these metrics will be investigated. Therefore, t copulas will be constructed for periods 1,2 and 3. To check whether there are indeed dynamics we will test H0 = φ1= φ2= φ3 where φi is the vector of

copula parameters in period i. Let us first consider copula model theory. Sklar (1959) states: “Let F be a joint distribution function with margins F1, ..., Fd. Then there exist a copula

C(·), continuous in [0, 1] such that:

F (x1, ..., xd) = C(F1(x1), ..., Fd(xd)). (5)

This implies that if F1, ..., Fd are univariate distribution functions, then the function F is a

joint distribution function with margins F1, ..., Fd.”

log-likelihood functions of the univariate margins are Ls(θs) = n X i=1 logfs(yi,s; θs), s = 1, .., 4, (6)

Furthermore, define the log-likelihood function of the joint distribution as L(θ, φ) =

n

X

i=1

logf (yi; θ; φ), (7)

where θ is the vector of parameters of the marginals, and φ that of the copula. Following the IFM method we first separately optimize the univariate models by maximizing equation (6). Define the obtained maximum likelihood estimates as ˜θ1, ..., ˜θ4. Then using these

esti-mates, L(˜θ1, ..., ˜θ4, φ) is maximized over φ. This is less efficient than estimating all parameters

θ1, ..., θ4, and φ jointly from equation (7), but is computationally less expensive. Therefore

it is worth comparing the asymptotic efficiency. Joe (1997) and Patton (2006b) provide ev-idence for consistency and asymptotic normality under regular conditions. Moreover, IFM allows for more flexible modeling techniques in the marginals, as the properties are deter-mined in a univariate manner. Hence the IFM method is used to find efficient parameters.

Using the IFM method we first have to construct decent marginals, as a misspecification error in ˜θs will result in an error in φ. Firstly, we will construct a univariate model for CAT

bonds. As discussed in Section 3, the histograms of T-Bonds and REIT show asymmetry. The high negative skewness of the CAT bonds and FTSE assets can be explained by extreme events, as Figures 11 and 12 show. For convenience we will consider both symmetric and asymmetric models. The high kurtosis suggests to use fat tail distributions. However distri-butions such as the t and hyperbolic distridistri-butions might not be fat enough, so we will also examine the use of extreme value theory to cover even more mass in the tails. To also describe the center of the density, the semi-parametric approach proposed by Danielsson and De Vries (2000) will be examined. They propose a Generalized Pareto Distribution (GPD) to model the tails, and an empirical distribution to capture the center. GPD focuses solely on extreme values, so using a threshold the left tail of the distribution can be modeled properly. Define the location, dispersion and shape parameters as µ ∈ R, σ ∈ R++ and ξ ∈ R respectively. The GPD is of the following form:

F_(ξ,µ,σ)(x) =  _{1 − 1 +}ξ(x−µ) σ
−1 ξ _{, for ξ 6= 0,} 1 − exp −x−µ_σ
, for ξ = 0, (8)

applied. A characterizing property of the GPD is that the mean excess function should grow linearly over threshold w, which can be used as a diagnostic to choose an appropriate threshold. For more details about the mean excess function see Appendix B. Define random variable X and the mean excess function as in (22). As depicted in Figure 15 in Appendix A, the mean excess function of the CAT bond losses starts to decrease linearly for x ≥ 0.70. Note that we are investigating losses, so we are regarding the left tail of the returns. Visual evidence is provided for a threshold of w = 0.70, which will therefore be pursued. Hence whenever the CAT bond decreases with more than 0.70%, this will be considered as an extreme, resulting in 14 extreme observations, enough to produce stable estimates. Using a QQ-plot we check whether the tail is properly described. The fact that Figure 15 shows a linearly decreasing function implies that shape parameter ξ is negative.

The T-Bond, FTSE and REIT assets do not show this structure of extreme values. The high kurtosis suggests to investigate t- and hyperbolic distributions. Both symmetric and asymmetric models will be examined, as some assets showed to be skewed. Parameters will be optimized using the Nelder-Mead method, as this is simple but efficient, and requires low storage (Ince and Kiranyaz (2014)). See Appendix B for a detailed description of this optimization method.

Suggested by Kole et al. (2007) a Student’s t copula will be used to join the marginal dis-tributions. They provide evidence showing that where the Gaussian and Gumbel copula over-and underestimate risk respectively, the Student’s t copula performs well. As an extensive copula investigation is beyond the scope of this thesis we assume similar results hold for our analysis. For convenience, performance is examined using the Cram´er-von Mises test. This is used to test the goodness-of-fit, as compared to the Kolmogorov-Smirnov method this is “almost invariably more powerful” (Genest et al. (1995)). A detailed description of this test can be found in Appendix A. Define T_νs−1 as the inverse of the Student’s t distribution of asset s with νs > 2 degrees of freedom. Define Gw as the SPD with threshold w. Furthermore,

define TR,ν as the 4 dimensional Student’s t distribution with correlation matrix R ∈ [−1, 1]4

and ν > 2 degrees of freedom. Then the t copula is given by:

C(u; R; ν) = TR,ν(G−1w (u1), Tν2−1(u2), Tν3−1(u3), Tν4−1(u4)), (9)

where usis a vector of the ordered probability transformations of asset s, and u = (u1, u2, u3, u4).

be estimated. The resulting ν will be considered fixed and is used as input variable in further analysis. The dependence parameters will again be estimated, but now using the previously obtained ν as arbitrarily given.

The conditional VaR represents the level of risk given that the other assets incur a high loss. Instead of a standard copula, which together with the marginals result in a joint prob-ability, a copula should be constructed such that the conditional probability is be obtained5. Therefore a conditional t copula will be constructed, i.e. C(u4|u1, u2, u3). This will use the

characteristics of the standard t copula constructed earlier, but add conditional constraints. In this analysis u1, u2 and u3 are the probabilities corresponding to the 95% VaR for each

marginal. Instead of producing joint probabilities, this copula will generate probabilities conditional on heavy losses of the other assets. Using the marginal distributions these prob-abilities can be transformed to quantiles, conditional on heavy losses of the other assets. These estimates will then be plotted against confidence level α ∈ (0, 1). The plots depict the sensitivity of an asset to market deviations. It represents the behavior of an asset, given that the other assets are in economic downturn. This metric, together with the dependence parameters, provide an answer to subquestion 1. To investigate the dynamics of this metric we construct a copula for the pre-crisis, crisis and post-crisis periods. Also these copulas will be transformed to conditional copulas and eventually plots of the conditional VaR will be compared to examine the dynamics of the metric. Defining a pre-crisis, crisis and post-crisis period and obtaining both metrics in those periods provides clear insight in the dynamics of the diversification benefits. It does however require proper specification of a crisis period, as misspecification could lead to false conclusions. Moreover it lacks the possibility to observe trends within a period. Therefore the analysis will be extended using a time-varying copula.

Time-varying t Copula

As discussed in Section 3 the volatilities of FTSE and REIT show to be unstable over time. Therefore a GARCH structure is used, such that time-varying marginals are generated. These marginals can be used to construct a time-varying correlation matrix, and subsequently a dynamic copula. Note that this model does assume a constant ν over time. Let us first however construct a static copula based on the time-varying marginals, such that insight is provided in the influence of the adjustment. As presented in Figure 14 the CAT bonds

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require lagged values when describing the conditional mean. This is supported by the Ljung-Box tests in Table 11. Moreover, to implement asymmetry in the volatility for some assets we will include a leverage effect, i.e. the volatility behaves differently in times of economic downturn than in times of prosperity. A positive shock has usually in these markets a different effect on volatility than a negative shock. To include this in our model a Threshold-GARCH (TGARCH) will be used instead of a standard GARCH. This will include an asymmetry variable η in the model. The number of lags is of the AR-, MA-, and GARCH part is based on the Akaike Information Criteria (AIC), where combinations until ARMA-GARCH(2,2,2,2) are considered for sake of parsimony. The optimal number of GARCH-lags show to be 1 for all variables, while only the CAT bonds show to improve adding 1 lag for the AR- and MA-part. let rs,t be the returns at time t for asset s. Then the conditional mean and variance

will be of the following form:

rs,t = µs+ α1,srs,t−1+ s,t+ α2,ss,t−1, (10) s,t= σs,tZs,t, (11) σ_s,tλ = ωs+ β1,s |s,t−1| − η1,ss,t−1
λ + β2,sσλs,t−1, (12) where Zt i.i.d

∼ SWN(0, 1). T-Bonds and RE show to be asymmetric. Therefore Z_t will follow a Skewed Student’s t distribution with dfs= νs. For the other assets Ztwill follow a symmetric

Student’s t distribution with dfs = νs. Note that for the FTSE, T-Bond, and REIT assets

the ARMA-part is insignificant, so α1 and α2 will be 0 for those variables. Furthermore,

the leverage effect is only appropriate for FTSE and REIT. A TGARCH is used to cover this, hence η1 6= 0 and λ = 2 for these assets. The volatility of CAT and government

bonds show less of these asymmetrical effects, hence a standard GARCH is used to model these volatilities. This implies η1 = 0 and λ = 1 for these assets. Furthermore, we have

the parameter restrictions ωs > 0, β1,s > 0, β2,s > 0, β1,s− η1,s > 0. These marginals can

subsequently be used to construct a time-varying t copula. While still preserving a fixed ν, correlation matrix R will consist a time-varying DCC-GARCH structure. Based on the AIC the number of lags is selected, where a DCC-GARCH(1,1) shows to fit best. Let the vector of inverse marginals be

Furthermore, consider the following variables: ¯

ρi,j := the unconditional correlation between i,t and j,t;

S := the unconditional correlation matrix of using elements ρi,j;

qi,j,t := the conditional covariance estimator;

Qt := the conditional covariance matrix, using elements qi,j,t;

ρi,j,t:= the correlation estimator;

Rt := the conditional correlation matrix, using elements ρi.j,t.

Then using a GARCH(1,1) structure in the conditional correlation matrix, the following holds:

qi,j,t= ¯ρi,j+ γ1(i,t−1j,t−1− ¯ρi,j) + γ2(qi,j,t−1− ¯ρi,j), (14)

where the condition γ1 + γ2 < 1 should hold for stability purposes. Rewriting this and

changing it to matrix notation we obtain:

Qt= (1 − γ1− γ2)S + γ1(ξt−1ξ0t−1) + γ2Qt−1, (15)

where Q0 = S. To find the correlation matrix Rt the following transformation is applied:

ρi,j,t=

qi,j,t

√

qi,i,tqj,j,t

. (16)

These estimators are used as elements in Rt, the time-varying input variable for our copula.

Rt will be a positive definite matrix, as Qt is a weighted average of a positive definite and a

positive semidefinite matrix, and equation (16) is a monotonic transformation (Engle (2002)). For sake of consistency, also a t copula will be used in this approach. The time-varying rank correlations will be plotted, such that the dynamics throughout the crisis can be examined.

5

Empirical Analysis

Pearson Correlation

For sake of consistency a similar crisis period will be defined as in Carayannopoulos and Perez (2015) and Cummins and Weiss (2009), which ranges from December 2007 to June 20096_.

To investigate the dynamics three subsets are defined: a pre-crisis period (January 2002 to December 2007), a crisis period (December 2007 to June 2009) and a post-crisis period (June 2009 to September 2017).

6

Table 2: Pearson’s Correlation Coefficients in each subset

Cat FTSE T-bond REIT

Set 1:Pre-crisis Cat 1.000 FTSE 0.027 1.000 T-Bond 0.001 −0.377∗∗∗ _1.000 RE 0.068 0.599∗∗∗ −0.099∗ _1.000 Set 2: Crisis Cat 1.000 FTSE 0.332∗∗∗ 1.000 T-Bond 0.126 −0.280∗∗∗ 1.000 RE 0.198∗ 0.880∗∗∗ −0.330∗∗∗ 1.000 Set 3: Post-crisis Cat 1.000 FTSE 0.034 1.000 T-Bond 0.062 −0.300∗∗∗ _1.000 RE 0.058 0.791∗∗∗ −0.002 1.000

The 1, 5 and 10 percent significance levels of the dependence coefficients are represented by∗∗∗,∗∗, and∗respectively

The correlation coefficients presented in Table 2 are qualitatively similar to those obtained by Cummins and Weiss (2009), and Carayannopoulos and Perez (2015). This shows validity of our dataset and similar conclusions can be drawn. In the periods before and after the crisis the correlation coefficients do not significantly differ from zero. This mean that in these periods CAT bonds can be regarded zero-beta investments. In the crisis period however,

H0 : ρ2,(cat,f tse) = 0 is rejected at a 1% significance level, implying that CAT bonds are not

a zero-beta investment anymore. Also the correlation between CAT and RE is significant at a 5% significance level. To investigate the dynamics we examine whether the increase in correlation is significant. The results are presented in Table 3.

Table 3: P-values of the Pearson correlation significance test

Periods CAT-FTSE CAT-Tbond CAT-REIT

1 - 2 0.01 0.30 0.27

2 - 3 0.01 0.58 0.22

1 -3 0.93 0.39 0.89

Periods 1,2 and 3 represent the pre-crisis, crisis and post-crisis period respectively

other assets is very low. Deterioration of other assets will therefore have limited effect on CAT bonds. The methodology of Cummins and Weiss (2009) is however only appropriate when investigating elliptical models. Zero correlation does not always imply independence between the assets. Therefore, a static copula is constructed to examine dependence. Static t copula

Using the IFM method we first have to construct decent marginals. Note that losses are modeled. Hence the right tail is from now on corresponding to losses instead of returns. As normality was rejected, alternative distributions should be examined with more mass in the tails such as the hyperbolic and t distribution. As mentioned in Section 3 the high skewness suggests to use asymmetric marginals. However, as discussed this high skewness is due to extreme events. The histograms of CAT bonds and FTSE suggest to use symmetric models. As these contradict we construct both for convenience and obtain the model selection information.

Table 4: Comparison of the univariate models

Gaussian t Model Hyperbolic

z }| {

ln L AIC z ln L }|ν AIC{ z ln L }| AIC{

Symmetric models CAT −374.60 753.21 334.60 2.00 −663.20 184.53 −363.07 FTSE −2054.74 4113.47 −1942.81 3.23 3889.61 −1948.63 3903.26 T-Bond −388.10 780.20 −380.06 9.67 766.12 −379.95 765.89 REIT −2220.22 4444.43 −2051.13 2.51 4108.27 −2065.67 4137.34 Asymmetric models CAT 334.60 2.00 −661.20 184.71 −361.43 FTSE −1941.40 3.27 3888.80 −1943.21 3894.42 T-bond −376.02 10.74 760.03 −375.38 755.76 REIT −2048.77 2.57 4105.54 −2062.51 4133.03

−3 −2 −1 0 1 2 −4 −2 0 2 4 Symmetic t distribution Theoretical Quantiles Sample Quantiles Symm t −1.0 −0.5 0.0 0.5 1.0 −4 −2 0 2 4 Hyperbolic distribution Theoretical Quantiles Sample Quantiles Symm hyp −1.0 −0.5 0.0 0.5 1.0 −4 −2 0 2 4 Gaussian distribution Theoretical Quantiles Sample Quantiles Gauss

Figure 5: QQ Plots for the CAT bond marginals

Figure 5 shows QQ-plots of the t-, hyperbolic and Gaussian distribution. Note that all models fit badly in the tails, as the underlying distribution shows heavier tails. As these models are incapable of modeling the tails, and especially the right tail, the Semi-Parametric Distribution (SPD) proposed by Danielsson and De Vries (2000) is examined. Furthermore, Table 4 shows the degrees of freedom ν corresponding to the t-distributions. T-distributions are bounded by a minimum ν of 2, which is also the value observed for the CAT bond. It might be that fatter tails are needed, but the t-distribution is unable to capture more mass in the tails. This confirms the choice of an alternative model to describe the CAT bonds. The figures depicting model fit of the SPD are presented in Figure 6. The QQ-plot of the right tail of SPD is depicted in Figure 6a. The CAT bond losses are better described using SPD, as the fit in the right tail is greatly enhanced using the GPD, which models the tails of the SPD. Figure 6b presents the overall density of the SPD. Even though the fit in the tail is greatly enhanced, it fails to describe the center properly. Still this approach is preferred above the t-distribution, as it is important to model losses properly when investigating the diversification benefits.

(a) QQ-plot of the right tail

−4 −2 0 2 4 0 1 2 3 Semi−Parametric PDF x Frequency

Pareto Lower Tail Kernel Smoothed Interior Pareto Upper Tail Observed Data (Kernel Estimate)

spd : GPD T

ail Fit

(b) Density plot SPD Figure 6: Figures representing model fit of the Semi-Parametric approach

tails of the distribution. Hence for the CAT, FTSE, T-Bond and REIT, we will use an SPD, symmetric t, and two asymmetric t distributions respectively. These marginals will be joined using a t copula. The degree of freedom ν is optimized using maximum likelihood, resulting in ν = 6. Repeating the process with a known df will result in the following dependency:

Table 5: Static t copula parameters based on static marginals

CAT FTSE T-Bond RE

CAT 1.000

FTSE 0.031 1.000

T-bond 0.068∗∗ −0.342∗∗∗ 1.000

RE 0.044 0.732∗∗∗ −0.081∗∗ 1.000

The 1, 5 and 10 percent significance levels of the dependence coeffi-cients are represented by∗∗∗,∗∗, and∗respectively

Table 5 presents the copula parameters φ, and shows that dependence between CAT bonds and FTSE and REIT is insignificant at a 5% level. However, there is significant dependence with T-bonds. CAT bonds can therefore not be regarded as zero-beta investments. However, as dependence is very low with all assets, they do tend towards zero-beta investments. To check validity of our copula the results will be compared with those obtained from an empirical copula, presented in Table 13 in Appendix A. The result are very similar, implying a good fit of our t copula. Also the Pearson correlations of the probability transformed data show to be similar (Table 14 in Appendix A), hence the rank correlations are well represented using a t copula. A more formal approach to test the fit of our copula is the Cram´er-von Mises goodness-of-fit test, described in Appendix A. This test uses the deviation of our hypothesized copula with the empirical copula to test copula fit. The results are presented in Table 15 in Appendix A. Using bootstrap and multiplier simulation methods a p-value is obtained of 0.362 and 0.382 respectively. This implies that the null hypothesis, stating that the data is consistent with our copula, cannot be rejected. The use of our t copula is therefore valid.

0 10 20 30 0.00 0.25 0.50 0.75 1.00 Confidence Level V aR CAT FTSE Real Estate Tbond

To graphically present the diversification benefits the conditional VaR is plotted against confidence level α ∈ (0, 1), depicted in Figure 7. Note that the losses are modeled in this paper, so the 95% quantile indicates a loss with an expected waiting time of 20 weeks. As can be seen, CAT bonds show limited dependence with the other assets. Conditional on high losses of the other assets there is still a probability of 0.20 of incurring no loss on the CAT bonds. This implies that the asset is not very sensitive to the state of other assets. Also when a loss is expected, there is still a very high probability that this will be moderate. Conditional on heavy losses of the other assets, there is still a 95% probability that the CAT bond losses will not exceed 3.02%. As the confidence level approaches 100% the conditional VaR of the CAT bond increases. Nonetheless the increase of the conditional VaR is evidently lower than those observed of the equity and RE assets.

Both the T- and CAT bond show to have diversification advantages. The difference be-tween the two can be explained using the discussion earlier about Figure 13. The probability of obtaining no loss is lower for T-bonds than for CAT bonds. Explanation for this is that a CAT bond rarely shows decreases in times of no natural disaster, while the government bonds shows to fluctuate more. However, as the confidence level approaches 100% government bonds show a more moderate increase of the conditional VaR than CAT bonds, as there are no extreme losses observed for government bonds, while for the CAT there are. One has to be careful though comparing assets within the plot, as each asset is conditional on a different set. Figure 7, but also the results from Table 5 provide an answer for our first subquestion. The CAT bonds show to have little to no dependence with the other assets and tend towards a zero-beta investment.

the Pearson correlation, and are even less when using an empirical copula. There might be two possible explanations for this. First of all this might be due to the fact that Pearson’s correlation only investigates linear relationships. In a non-elliptical setting, dependence is not fully described using linear correlations, hence differences are observed. Furthermore, as the copula uses rank correlations, the effect of outliers, or extreme events is lower. Lastly, an explanation might be that we are still unable to fully describe CAT bonds using our marginal model. A misspecification error will lead to an error in the dependence parameters.

Table 6: Copula Parameters in subsets

Cat FTSE T-bond RE

Set 1:Pre-crisis Cat 1.000 FTSE 0.064 1.000 T-Bond 0.028 −0.368∗∗∗ 1.000 RE 0.102∗ 0.592∗∗∗ −0.081 1.000 Set 2: Crisis Cat 1.000 FTSE 0.186∗ 1.000 T-Bond 0.165 −0.362∗∗∗ 1.000 RE 0.077 0.864∗∗∗ −0.391∗∗∗ 1.000 Set 3: Post-crisis Cat 1.000 FTSE −0.011 1.000 T-Bond 0.083∗ −0.293∗∗∗ _1.000 RE 0.018 0.768∗∗∗ 0.042 1.000

The 1, 5 and 10 percent significance levels of the dependence coefficients are represented by∗∗∗,∗∗, and∗respectively. The ν of subsets 1,2, and 3 are respectively 24, 6, and 5.

To formally test the dynamics of the parameters we use a likelihood ratio test, where we test H0: φ1 = φ2 = φ3 against H1 : H0 not true. Using the log-likelihoods of the constructed

copulas a test statistic of 24.21 is obtained, which is higher than χ2_14,0.95 = 23.687. This implies that we can reject H0 at a 5% confidence level and hence the parameters do differ

significantly between periods. We conclude that there is a significant difference in dependence between crisis and non-crisis periods8. As the dependence of CAT bonds with the other assets increases during the crisis period, the diversification benefits are less. Despite the benefits are less, the dependence parameters are still very low, even insignificant at a 5% level,. This suggests that also in crisis periods the benefits are present. To graphically show the diversification benefits the conditional VaR is plotted against its confidence level. The plots

7

See Appendix A for the details of this test. 8

of each period are presented in Figure 8. 0 10 20 30 0.00 0.25 0.50 0.75 1.00 Confidence Level V aR CAT FTSE Real Estate Tbond Pre−crisis period (2002/01/04 − 2007/12/01) 0 10 20 30 0.00 0.25 0.50 0.75 1.00 Confidence Level V aR CAT FTSE Real Estate Tbond Crisis period (2007/12/01 − 2009/06/01) 0 10 20 30 0.00 0.25 0.50 0.75 1.00 Confidence Level V aR CAT FTSE Real Estate Tbond Post−crisis period (2009/06/01 − 2017/08/01)

Figure 8: Conditional VaR behavior in each period

of the swap. One of the counterparties to issue these TRS’s was Lehman Brothers, hence the collapse left the assets vulnerable. Furthermore, the quality of the invested assets were not as high as initially expected. Often the portfolios included mortgage backed securities. While these assets were often AAA or AA+ rated these securities showed to be very insecure and crashed during the economic crisis. As Lehman Brothers was unable to deliver the TRS, this influenced the returns of the CAT bonds and hence its price. As the assets were assumed to be very safe, little attention was paid to the composition of the portfolio and hence no further hedging strategies were applied. Often corporate bonds were included in the portfolio, which crashed as no hedging strategy was applied. All of this resulted in less diversification benefits as initially expected9.

Another reason for higher expected losses during the crisis period is the fact that CAT bonds are traded mark-to-market. The demand for money is higher in times of economic downturn, hence investors will more often liquidize their assets. As CAT bonds are freely tradable, part of the investors will liquidize these bonds to obtain a well balanced asset-liability structure. Hence dependence will always slightly increase during a financial crisis.

After the crisis a reversible effect is noted. The level of the losses converges again to its old state, where moderate losses are expected. Conditional on heavy losses of the other assets, there is a probability of 0.95 that CAT bond losses will not exceed 2.89%. The frequency of a loss did however not recover, as there is still a 90% probability of incurring a loss on CAT bonds. This suggests that the aftermath of the financial crisis is covered in the post-crisis period, and not the crisis period. As the results did not show to be very robust we will disregard the need for specification of a crisis period using a dynamic t copula.

Dynamic t copula

Besides the copula, also the marginals are investigated in a dynamic setting. To cover the unstable volatility, time-varying marginals are constructed. Based on the analysis in the previous subsection, the marginals use a t-distribution, symmetric for CAT bonds and FTSE, and asymmetric for T-Bonds and REIT. The volatilities are described using GARCH and TGARCH approaches. The parameters of these models are presented in Table 16 in Appendix A. As can be seen nearly all variables show to be significant at a 1% level. This implies that

9_{See discussion in Carayannopoulos and Perez (2015). For a more details see Krutov (2010) and Watson}

each variable used is descriptive, hence the choice of our models is valid. Note that β2 is

significant for each asset, suggesting the importance of using a GARCH instead of an ARCH model. Also the leverage variables, represented by η1 show to be significant at a 10% level,

implying that the TGARCH is well described. The df of the marginals are comparable to those obtained in the static approach. Using these marginals we will first construct a linear copula, resulting in the following copula parameters:

Table 7: Static t copula parameters based on dynamic marginals

CAT FTSE T-Bond RE

CAT 1.000

FTSE 0.005 1.000

T-bond 0.057 −0.360∗∗∗ 1.000

RE 0.012 0.990∗∗∗ −0.260∗∗∗ 1.000

The 1, 5 and 10 percent significance levels of the dependence coeffi-cients are represented by∗∗∗,∗∗, and∗respectively. ν = 12.

Comparing the results of Tables 5 and 7 shows that including a time-varying volatility has only limited influence on the dependence of CAT bonds with other assets. The correspond-ing parameters are insignificant at a 5% confidence level, hence CAT bonds can be regarded zero-beta investments. To investigate the dynamics a time-varying t copula is constructed, where instead of using a constant R a DCC-GARCH(1,1) structure is used to construct a time dependent correlation matrix. The parameters are presented in the last column of Table 16. Both parameters show significance at a 1% level, which confirms our choice of a DCC-GARCH structure. This is supported by an increasing AIC when not including the lags (AIC = 6.892 against AIC = 7.006). As γ1 is close to 0 and γ2 is close to 1 it can be concluded

that the present correlation is mostly described by that of last week. Hence the differences with the static approach will be limited. While the main results are similar, dynamics can be investigated without specifying a crisis period. Plotting the conditional correlations over time, as depicted in Figure 9 provides interesting insights.

−0.50 −0.25 0.00 0.25 0.50 2003 2005 2007 2009 2011 2013 2015 2017 Date Correlation CAT−FTSE −0.50 −0.25 0.00 0.25 0.50 2003 2005 2007 2009 2011 2013 2015 2017 Date Correlation CAT−Tbond −0.50 −0.25 0.00 0.25 0.50 2003 2005 2007 2009 2011 2013 2015 2017 Date Correlation CAT−REIT

Figure 9: Dependence of the CAT bond with the other assets over time.

not influenced by the economic state itself. The financial crisis, as stated by the National Bureau of Economic Research, started in December 2007, while the CAT bonds were affected 9 months later. One explanation for this could be Ike, a hurricane occurring in September 2008. The black dotted vertical lines represent US hurricanes with corresponding insured losses exceeding 10bn dollars. As can be seen the correlations increase right after such an event. Noy (2009) provides evidence that a natural disaster has a negative effect on the economic state, hence the FTSE, T-bond and REIT indices decrease. Also CAT bond prices will decline whenever a big catastrophe approaches. Investors are afraid the insured losses will exceed the trigger threshold and their money shall be used a coverage. As CAT bonds are mark-to market, investors will sell these bonds and prices decline. Combining these trends results in an increasing correlation.

covered by a TRS. With the collapse of Lehman however, one of the TRS counterparties, this changed. This effect is enlarged by the downgrading of Lehman bonds.

During the aftermath of the crisis slight correlation between the CAT and equity market is still observed. As both markets are resurrecting after the market a common trend is observed, hence dependence increases. Events like the Flash Crash in 2010 and the stock market Crash in 2011, due to downgrading of the US credit rating, heavily influenced the equity market. While these assets deteriorated, CAT bonds remain unaffected, resulting in a decreasing correlation. Similar analysis can be applied to the dependence between CAT bonds and real estate. During the crisis dependence increased, but the rebounds in 2010 and 2011 returned correlation back to the pre-crisis level.

In contrast to the equity and real estate assets, the CAT and T-bonds showed to become more dependent in recent years. In 2013 the European Central Bank (ECB) initiated the Quantitative Easing (QE) program10. As government bonds are heavily purchased, bond prices increase. Also CAT bonds prices steadily increase, as very few mayor catastrophes occurred. As similar trends are observed correlation increases. A noteworthy observation is the increase in July 2011. CAT bond prices increase as the Japanese Tsunami just passed in March earlier that year. This resulted in 38 billion insured losses, hence part of the collateral account is liquidated. To recharge insurance companies increase the premium to attract in-vestors. This implies rising CAT bonds prices. Later that year also government bond prices increased heavily. In July 2011 the US started its QE program, hence buying huge amounts of US government bonds. As both assets showed a strong increase dependence increases in 2011.

Using a time-varying copula has already showed its advantage, as trends can be analyzed. However, also the convenience of no period specifications is visible in Figure 9. Evidence is provided that it wasn’t the financial crisis on itself that caused the correlation increase, but the combination with the collapse of Lehman Brothers. This supports our answer to the second subquestion. CAT bonds showed not to be immune to a financial crisis, as correlations increased due to the collapse of Lehman. Despite the increase, the correlations do remain low, hence diversification benefits are still present during a financial crisis.

10

6

Conclusion

This paper illustrates an effective approach, using a copula framework, to examine the di-versification benefits of CAT bonds in a portfolio consisting CAT bonds, equity, real estate and fixed income. To investigate the diversification benefits two subquestions are defined. The first questions whether CAT bonds are a zero-beta investment, i.e. show no dependence with the other assets. To investigate this two metrics are proposed. First of all, the copula parameters provide strong evidence that CAT bonds show little to no dependence with the other assets. Furthermore, the conditional VaR shows that even in times of extreme losses, there is still a high probability that CAT bonds will not incur a loss. Also when incurring a loss, there is a high probability that this will be moderate. Our second subquestion regards the dynamics of the diversification benefits. Comparison of the copula parameters of a pre-crisis, crisis and post-crisis period shows that the dependence of CAT bonds with other assets increases only moderately in times of crisis. Investigating whether the parameter change is substantial does not provide a robust result, as the outcome is highly dependent on the crisis period chosen. Constructing the conditional VaR provides strong evidence that diversifica-tion benefits are still present during the financial crisis, as the condidiversifica-tional VaR show to be less affected by the financial crisis than that of equity and RE.

We do note that CAT bonds are affected by the crisis. Prior beliefs state that CAT bonds are isolated from market risk as their structure should provide them to be exposed to the risk of a catastrophe only. Our analysis provides evidence against this statement, as both metrics show to be affected by the financial crisis. The sensitivity to the economic state can be explained by the crash of Lehman Brothers in September 2008. This event exposed sev-eral vulnerabilities within the structure of CAT bonds. Hedging the assets in the collatsev-eral account using a total return swap was shown to be ineffective as the counterparty could still default. Furthermore the assets in the account demonstrated to be of lower quality than initially expected. This resulted in an increasing dependence with other assets during the financial crisis in 2008.

A disadvantage of this method is the need to specify a crisis period. The results of the likelihood ratio test, testing H0 : φ1 = φ2 = φ3is highly dependent on the crisis period chosen

throughout the financial crisis without the specification of a crisis period. As correlation shows to be heavily determined by the correlation of last week, the main results will be similar to those obtained using a static copula. This approach does however provide more insight in the cause of the increasing correlation, as the conditional correlation increases in September 2008 while deterioration of the equity and real estate assets started months earlier. Hence evidence is provided for our explanation, stating that CAT bonds were vulnerable to market influences due to weaknesses in their structure. An answer is provided for our second subquestion: Diversification benefits decrease in times of crisis, as CAT bonds do show to be affected by economic downturn. We do note that even though the benefits are less, they are still present, as compared to equity and real estate the increase in dependence is inferior.

An important remark is that the level of dependence is evidently lower during the crisis than Pearson’s correlation represents. Reasoning for this might be the use of rank correlations instead of examining linear relationships. Furthermore, Pearson’s correlation representative for dependence if distributions of the assets are of the elliptical class. When this is not true, a copula framework has shown to be effective. When modeling CAT bonds, fat tailed distribu-tions such as t or hyperbolic distribution show to be unable describe the losses properly. To enhance fit in the tail, extreme value theory is used. This theory is however not applicable for T-bonds, as extremes are rarely observed. As each asset can be modeled individually using a copula, misspecification errors are minimized and dependence can be investigated properly.

Combining both subquestions we can conclude that CAT bonds are a great source of diversification. The bonds tend towards zero-beta investments, as very low dependence is observed. We do note that within crisis periods CAT bonds show slight sensitivity to market influences. The conditional VaR however shows that even in these periods the expected losses are moderate, hence in times of economic downturn diversification benefits are present.

7

Discussion

the fit in the tails. However this approach does require definition a threshold such that sta-ble parameters are obtained while the bias due to misspecification of extremes is minimized. As CAT bond data is provided on a weekly basis the condition to obtain stable parameters might have induced a bias. Using a Log Phase-type distribution disregards this inefficiency (Ahn, Kim, and Ramaswami (2012)). An alternative is be the use of time-varying jump tails, proposed by Bollerslev and Todorov (2014). Using these models might reduce the misspeci-fication error and hence better describe dependence. These models are beyond the scope of this paper, but suggested for future analysis.

One of the main conclusions relates to the dynamics of the diversification benefits. During the crisis in 2008, CAT bonds were exposed to some weaknesses in their structure, resulting in an increasing dependence with other assets. We can however not conclude that a similar pattern will be observed in future crises. After noting the vulnerability, several changes are implemented in the structure of CAT bonds. First of all, more attention is paid to the compo-sition of the portfolio. Assets such as corporate bonds (Lehman Bonds) and mortgage-backed securities are not considered as valid investments. Nowadays money in the collateral account is invested in money market funds11. These investments have a lower yield, but provide more safety than the portfolio used before. As often the account is hedged within the money market fund, there is no more need for a TRS. Using this new structure, the dependence to market changes should be limited.

Another reason that CAT bonds might be less affected by market influences is that our analysis only considers publicly traded bonds. In periods of economic downturn the demand for money is higher. Mark-to market trading eases the liquidation process, resulting in a structural increase in correlation with other assets during a crisis. For future analysis it might be interesting to include bonds that are traded over the counter12. Comparison with our results will provide insight in the magnitude of the effect of an easier liquidation process.

Lastly we note that the time-varying t copula uses a fixed ν, while the parameters in the likelihood ratio test did not show to be constant. An assumption made in the analysis is that the df is constant, and the dependence parameters vary over time. Whether this is a valid assumption is debatable, which has to be acknowledged when interpreting the results.

11

Big funds consisting government bonds only, and with short maturities. 12

8

References

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9

Appendices

9.1 Appendix A

Table 8: Variable descriptions

Variable Description

BB-rated CAT bond Index Provided by Swiss Re. Tracks the total returns for all the outstanding USD denominated CAT bonds. FTSE All World Index A free float market cap weighted index representing

performance of large & mid cap stocks from both Developed and Emerging markets

Barclays Global Treasury Index A Market Value weighted index representing total returns in USD, hedged against currency movements. Contains issues from 37 countries, both Developed and Emerging markets

S&P Global REIT Index A free float market cap weighted index representing the total returns of Real Estate Investment Trusts. All publicly traded equity REITs are included in both Developed & Emerging Markets

Table 9: Mayor Global Catastrophes

Date Type Name Economic Loss Insured Loss Victims Country

02-05-2003 Storm 5.59 4.26 51 US

11-08-2004 Hurricane Charley 23.33 10.23 36 US

26-08-2004 Hurricane Frances 13.5 6.52 50 US

02-09-2004 Hurricane Ivan 28.51 16.75∗ 119 US

06-09-2004 Typhoon Songda 9.26 4.65 45 Japan

13-09-2004 Hurricane Jeanne 10.37 4.99 3034 US

26-12-2004 Tsunami 16.85 2.59 220000 Indonesia

25-08-2005 Hurricane Katrina 175.51 82.31∗ 1836 US

20-09-2005 Hurricane Rita 20.06 13.46∗ 34 US

19-10-2005 Hurricane Wilma 25.16 15.76∗ 53 US

18-01-2007 Storm Kyrill 11.81 7.2 54 Germany

06-09-2008 Hurricane Ike 42.72 23.03∗ 193 US

27-02-2010 Tsunami 33.68 8.98 562 Chile

04-09-2010 Earthquake 6.96 5.61 100 New Zealand

22-02-2011 Earthquake 21.77 17.41∗ 185 New Zealand

11-03-2011 Tsunami 228.55 38.09∗ 18451 Japan 27-07-2011 Flood 50.06 16.32∗ 815 Thailand 15-07-2012 Drought 15.99 11.73∗ 123 US 24-10-2012 Hurricane Sandy 74.6 30.74∗ 237 US 27-05-2013 Flood 7.38 4.34 25 Germany 27-07-2013 Hail 5.07 4.03 0 Germany 14-04-2016 Earthquake 27.54 4.98 137 US 12-24-2016 Hurricane Matthew 12.24 4.11 734 US

Losses are displayed in billions of US dollar

CAT returns Frequency −4 −2 0 2 4 0 50 100 150

FTSE returns Frequency −20 −15 −10 −5 0 5 10 0 20 40 60 80 100 120 REIT returns Frequency −15 −10 −5 0 5 10 15 0 20 40 60 80 100 140

Figure 12: Histograms of FTSE and REIT

−5.0 −2.5 0.0 2.5 5.0 2003 2005 2007 2009 2011 2013 2015 2017 Date Retur ns CAT −5.0 −2.5 0.0 2.5 5.0 2003 2005 2007 2009 2011 2013 2015 2017 Date Retur ns T−Bond

The Jarque-Bera test is a goodness-of-fit test for testing: H0: data follows a normal distribution.

Ha: data follows an alternative distribution.

Define n as the number of observations, S and C as the sample skewness and kurtosis respec-tively and let k be the number of regressors. Then the following test statistic is used:

J B = n − k + 1 6  S2+1 4(C − 3) 2 

This will be tested against a chi-squared distribution with df = 2. Normality implies S=0 and C=3. As can be seen any deviations result in an increasing statistic. The following results are obtained:

Table 10: Jarque-Bera Test Results

Variable JB p-value CAT 132260 0.000 FTSE 6910.2 0.000 T-Bond 41.7 0.000 Real Estate 4504.6 0.000 0 10 20 30 0.0 0.6 Lag A CF CAT 0 10 20 30 0.0 0.6 Lag A CF FTSE 0 10 20 30 0.0 0.6 Lag A CF Bond 0 10 20 30 0.0 0.6 Lag A CF Real Estate 0 10 20 30 −0.05 0.10 Lag P ar tial A CF CAT 0 10 20 30 −0.05 0.10 Lag P ar tial A CF FTSE 0 10 20 30 −0.05 0.05 Lag P ar tial A CF Bond 0 10 20 30 −0.05 0.10 Lag P ar tial A CF Real EState

The Ljung-Box test is a test to examine the population autocorrelations, used when fitting an ARIMA-model. Define the autocorrelations as ρ. Then:

H0: ρ1= ρ2 = ... = ρp = 0

Ha: ρi 6= 0.

Define sample size n and maximum number of lags as p. Then the following test statistic is used: LB = n(n + 2) h X j=1 ˆ ρ2_n,j n − j

This should follow a chi-squared distribution with df=p. the following results are obtained:

Table 11: Ljung-Box Test Results

Variable LB p-value CAT 56.168 0.000 FTSE 17.085 0.004 T-Bond 4.339 0.500 Real Estate 12.439 0.029 −4 −3 −2 −1 0 1 2 0 1 2 3 4

Mean Residual Life Plot

w

Mean Excess

−15 −10 −5 0 5 10 15 −10 −5 0 5 10 15 20 Symmetic t distribution Theoretical Quantiles Sample Quantiles Symm t −10 −5 0 5 10 −10 −5 0 5 10 15 20 Hyperbolic distribution Theoretical Quantiles Sample Quantiles Symm hyp −5 0 5 −10 −5 0 5 10 15 20 Gaussian distribution Theoretical Quantiles Sample Quantiles Gauss −10 −5 0 5 10 15 20 −10 −5 0 5 10 15 20 Asym. t distribution Theoretical Quantiles Sample Quantiles Asymm t −5 0 5 10 −10 −5 0 5 10 15 20

Asym. Hyperbolic distribution

Theoretical Quantiles

Sample Quantiles

Asymm hyp

Figure 16: QQ-Plots regarding FTSE models

−1.0 −0.5 0.0 0.5 1.0 1.5 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 Asym. t distribution Theoretical Quantiles Sample Quantiles Asymm t −1.0 −0.5 0.0 0.5 1.0 1.5 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

Asym. Hyperbolic distribution

Theoretical Quantiles Sample Quantiles Asymm hyp −1.0 −0.5 0.0 0.5 1.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 Gaussian distribution Theoretical Quantiles Sample Quantiles Gauss