Diversication Benets of Cat Bonds:

An In-Depth Examination∗

Karl Demers-Belanger & Van Son Lai †

July 15, 2019

∗We would like to thank the Fonds Conrad Leblanc, the Québec Autorité des Marchés Financiers, andthe Social Sciences Research Council of Canada for their nancial support. We thank S. Chrétien for hisvaluable comments.†Laval University, Faculty of Business Administration, Quebec (Quebec), Canada G1V 0A6 & IPAG

Business School, Paris, France; E-mail: karl.demers-belanger.1@ulaval.ca & vanson.lai@fsa.ulaval.ca.

Diversication Benets of Cat Bonds:

An In-Depth Examination

Abstract

We investigate whether the inclusion of Cat Bonds in portfolios composed of tra-

ditional assets and common factors is benecial to investors for the period of 2002 to

2017. Various mean-variance spanning tests show that under dierent market condi-

tions, the addition of Cat Bonds gives rise to previously unattainable portfolios. Using

the Engle (2002) Dynamic Conditional Correlation (DCC) model, we nd that includ-

ing Cat bonds increases signicantly the time-varying Sharpe ratio and the Choueifaty

and Coignard (2008) diversication ratio. Cat Bonds provide needed diversication

during critical times particularly during episodes of crisis and of high volatility. Under

the second-order stochastic dominance eciency (SDE) tests, the null hypothesis that

portfolios without Cat Bonds are ecient cannot be rejected. Out-of-sample analyses

indicate that the performance of portfolios with Cat Bonds included varies depending

on the performance measures employed, the portfolio construction techniques used and

the assets or factors considered.

Keywords: Catastrophe Bonds, Asset Allocation, Factor Investing, Diversication, Stochas-

tic Dominance Eciency, Mean-Variance Spanning, Portfolio Optimization, Time Vary-

ing, Regime Switching, Dynamic Correlation.

1 Introduction

With average annual temperatures rising and frequencies of natural disasters increasing in allparts of the world, dire environmental impacts, vital socio-political and important economicproblems associated to global warming are growingly alarming. In its 2007 report (Bernsteinet al., 2008), the Intergovernmental Panel on Climate Change (IPCC) describes the eectsof global warming that are already being felt and the IPCC expects these eects to grow inmagnitude and costs in the future.1 Catastrophic risks associated with these environmentalphenomenon are of great importance for nancial markets and institutions, all quarters andconstituencies as well as governments whose stakes are greatly aected by natural disasters.

It ensues that increases in the frequency and severity of natural catastrophes in recent yearshave propelled the use of alternative risk-transfer instruments for managing catastrophicrisks. To cover against disaster risk and transfer some of the risks they do not want to retain,insurance and reinsurance companies typically use reinsurance or retrocession (reinsurancefor reinsurers). However, for catastrophic natural disasters, the margins charged by reinsur-ers are often prohibitive. Therefore, insurance-linked securities (ILS) have been launched tocover these types of losses at relatively lower costs. Among these alternative risk transferinstruments, the most important asset class is catastrophe bonds (Cat Bonds hereafter), seeCummins (2008), Bouriaux and MacMinn (2009), Barrieu and Albertini (2010), Cummins(2012), Smack (2016). The size of the outstanding catastrophe bond and insurance-linkedsecurities markets at the end of 2018 is $37.8 billion.2 Naturally, Cat Bonds returns incorpo-rate risk premiums rewarding investors who are willing to assume a hardly diversiable andhedgeable risk. In fact, among ILS, Cat Bonds are the only nancial instrument securitizedand traded in secondary markets. As a complement to reinsurance, Cat Bonds have beenused to transfer the risk attached to the highest layers of reinsurance. While both reinsuranceand Cat Bonds oer companies a means to transfer disaster risk, only Cat Bonds use thecapital markets for this purpose, see e.g., Canter et al. (1996), Krutov (2010), Kish (2016).

There are two main benets from using Cat Bonds. First, their presumed zero or very lowcorrelation with other nancial assets allows for additional diversication to a portfolio, e.g.,Litzenberger et al. (1996), Hoyt and McCullough (1999), Kish (2016), Sterge and van derStichele (2016), and second, their historical risk-adjusted returns are attractive to investors,see e.g., Schöchlin (2002), Kusche (2013) among others. Cat Bonds are considered as ahigh-yielding xed income asset class with returns independent from macroeconomic risksand cycles. In addition, Cat Bonds show a high level of intra-class diversication providedby their correlation to dierent and independent risk factors arising in dierent parts of theplanet (i.e., tsunami, hurricanes, oods, etc.). Hence, Cat Bonds, exhibiting a low correlationto other traditional asset classes (e.g., equities, bonds and commodities) oer institutional

1In 2018, the IPCC sounds an alarm on "the impacts of global warming of 1.5 degrees Celsius abovepre-industrial levels and related global greenhouse gas emission pathways, in the context of strengthening theglobal response to the threat of climate change, sustainable development, and eorts to eradicate poverty",Special Report on Global Warming of 1.5oC at https://www.ipcc.ch/.

2See Q4 2018 Catastrophe Bond & ILS Market Report at http://www.artemis.bm and Insurance-Linked Securities market update, Swiss Re, August 2018 at https://www.swissre.com.

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investors a great portfolio diversication with an appealing risk-return prole. Institutionalinvestors (mainly pension funds and hedge funds) looking for a steady, relatively high-yieldand exotic asset class in the current low-interest rate environment are said to be lining upcapital to support the increased appetite in the insurance-linked securities market for CatBonds (see e.g., Carhart et al. (2014), Sterge and van der Stichele (2016), and Kish (2016)).

The purpose of this paper is to examine in-depth whether the addition of Cat Bonds to aninvestor portfolio does eectively provide him the benets of diversication. We conduct ananalysis of the time-varying performance associated with the inclusion of Cat Bonds underdierent nancial market conditions over the period of 2002 to 2017. We consider two dierentbenchmark universes. The rst is formed from traditional asset classes and the second onefrom common factors. Well-known factors in the literature which were constructed as theunderlying drivers of risk and return across assets and asset classes, can be macro oriented(e.g., economic growth, ination) or style oriented (e.g., value, momentum, quality). Further,factors are not directly investable, but factor exposures are an oshoot of investing in assets.With relatively more stable returns than those from asset class returns over time, dependingon the market environment some factors perform better than others, see for instance, Ang(2014), Naik et al. (2016), Dimson et al. (2017), Cerniglia and Fabozzi (2018).

The asset classes are U.S. equities, global equities ex-U.S., emerging markets equities, real-estate, U.S. treasury bonds, U.S. corporate bonds, U.S. high-yield bonds and commodities.The factors consist of equity market, value, size, momentum, volatility, mortgage, default,term, high-yield and commodity curves. To represent Cat Bonds, we use ve dierent SwissReinsurance indexes.3

Standing out from previous studies, we conduct our study using the following four dier-ent approaches. First, we perform a battery of mean-variance spanning tests based on themethodology developed in Kan and Zhou (2012). Furthermore, to split the sample into twoeconomic regimes, we use three approaches 1- the separation called by the NBER data, 2-the turbulence index à la Kritzman and Li (2010), and 3- a Markov-Switching model appliedto the US stock market index and the US bond market index à la Hardy (2001). We conductour tests using the full sample as well as the regime-based periods. We nd that the additionof Cat Bonds leads to portfolios previously unattainable regardless of the regimes.

Second, by means of the Engle (2002) dynamic conditional correlation model (DCC), westudy the time-varying eects of including Cat Bonds in constructing portfolios. We start byestimating the correlations between Cat Bonds and other assets. We then use these correla-tions to obtain the maximum Sharpe ratio and maximum diversication à la Choueifaty andCoignard (2008) portfolios with and without Cat Bonds. Portfolios with Cat Bonds yieldhigher Sharpe ratios and diversication ratios than those without these.

3Weekly prices from January 4, 2002 to June 30, 2017 are used. Albeit the Swiss Re indexes are notinvestable, Jaeger et al. (2011) present a methodology that would allow the construction of a Cat Bondsindex that would be investible. Their procedure is implementable and enables investors to have exposure toCat Bonds returns. In fact, the index created presents descriptive statistics similar to the Swiss Re indexesand exhibits a very high correlation with these. The growing popularity of Cat Bonds might lead to a launchof Cat Bonds Exchange Traded Funds (ETFs), facilitating investors' purchase of Cat Bonds.

2

Third, to determine whether a portfolio of traditional assets stochastically dominates a port-folio created from the same universe with Cat Bonds added, we employ a two-step methodas in Daskalaki et al. (2017) to perform stochastic dominance eciency tests à la Scaillet andTopaloglou (2010).4 We run the tests on the full sample under the dierent market condi-tions mentioned above. Based on the second-order stochastic dominance criterion, we ndthat the null hypothesis that portfolios that do not include Cat Bonds are ecient cannotbe rejected. This means that adding Cat Bonds might not achieve diversication when weconsider higher moments of the return distribution.

Finally, by means of a rolling window method, we analyze the out-of-sample performance ofthe portfolios of maximum Sharpe ratio, of maximum diversication and those constructedfrom the second-order dominance criterion. To take into account the non-normal distribu-tion of returns and the portfolios' turnovers, we use the conditional Sharpe ratio (CSR) àla Maillard (2018) and the Omega ratio à la Keating and Shadwick (2002) as metrics ofperformance. The results dier depending on the performance measure used, the portfolioconstruction technique employed to obtain the optimal portfolio and the assets or factors uni-verse considered. In this context, we conclude that adding Cat Bonds is not always desirablefor investors.

We make a number of contributions to the current Cat Bond literature. To consider thewhole distribution of Cat Bond returns, we use the second-order stochastic dominance e-ciency (SSDE) criterion. In addition to studying the dynamic aspect of the correlation, wealso quantify the variation over time of the eects of including Cat Bonds in a portfolio.Furthermore, we also consider a universe of factors instead of simply asset classes. Finally,we use out-of-sample performance measures which allow for deviations from normality. Tothe best of our knowledge, we are the rst, not only to use both mean-variance spanning andstochastic dominance approaches, but also to take into account dierent market conditionsto study the diversication eects of Cat Bonds in a framework of portfolios constituted ofboth asset classes and factors.

The rest of the paper is organized as follows. We provide a review of the literature in Section2. In Section 3, we detail our methodological approaches and in Section 4, we describe ourdata. After revealing the results in Section 5, we conclude in Section 6.

2 Literature Review

Catastrophe bonds work like traditional ones, i.e., a Cat Bond pays coupons at a regularinterval and a notional amount at maturity. A major dierence with the usual bond case isthat these payments are made contingent to the nonoccurrence of a natural disaster. Thedenition of disaster is specied in the contract and can take many forms, see for instance,Cummins (2008), Barrieu and Albertini (2010), Cummins (2012), Smack (2016). For a recentand concise primer and retrospective on Cat Bonds, see Polacek et al. (2018). In the risktransfer, the reinsurance or insurance company, called the sponsor, does not issue the bond

4For reasons described later, we only employ the second-order stochastic dominance (SSDE).

3

directly to the market. This issuance is made via a special-purpose vehicle (SPV) which isgenerally located in a tax haven. The rm enters into a reinsurance contract with the SPVwhich issues the bond to the market. The sponsor credit risk is hence removed from theinstrument. The SPV then invests the issue proceeds in low-risk and short-term instrumentssuch as treasury bonds or AAA corporate bonds kept in a collateral trust account. To changetheir xed incomes into variable ones, most Cat Bonds take a position in a total return swap(TRS). During the life of the bond, the SPV pays coupons at a oating rate, generallyLIBOR plus a risk premium. If the disaster occurs, the coupon payment is suspended and,depending on the severity of the damage, the investor may lose the principal amount, seee.g., Carayannopoulos and Perez (2015).5

It is important to note that Cat Bonds are generally rated between BB and B and shouldbe compared with high-yield bonds in terms of risk-adjusted performance. As mentionedabove, Cat Bonds do oer attractive risk-return opportunities. Schöchlin (2002) shows thatbefore 1998, the Cat Bonds' Sharpe ratio exceeded the one obtained from bonds with thesame rating. She also notes that in 2000, the average return at maturity of the Cat Bondswas 100-200 basis points above their actuarial value. He further documents that in the late2000s, returns from Cat Bonds and those from corporate bonds with similar ratings convergeto the same value. In sum, compared to stocks and government bonds, Cat Bond returnsappear to be more attractive. However, compared to high-yield bonds, they exhibit a similarrisk-adjusted performance.6 Since the 2008 crisis, investors have been looking for assets thatgenerate attractive returns in a low interest rate environment. Furthermore, investors areincreasingly aware of the importance of not only diversication but also the much soughtafter assets that are weakly correlated with the nancial markets. Therefore, attractivereturns and low correlations with the other markets (nancial and alternative assets) drivethe increasing popularity of the Cat Bonds market.7 Cat Bonds yields reached a level of 2.5%

5There are dierent types of triggers specied in the contract, (Goyal and Jain, 2014):

1. Indemnity: According to the issuer's losses.

2. Modelled losses: Calculated from a model provided by a risk modeling rm.

3. Industry index: Losses associated with an entire sector reported by PCS (Property Claim Services) orPERILS.

4. Parametric: Triggered by natural hazards parameters, for example, wind speed, Richter scale, etc.

5. Hybrid: Combination of several triggers.

6Krutov (2010) among others, presents the causes of the excess returns over stocks and bonds:

1. Novelty premium: Few investors are familiar with the instrument.

2. Liquidity premium: Cat Bonds are not traded frequently. They are generally held until maturity.

3. Sudden death premium: Investors could lose all in the event of a major disaster.

4. Asymmetric information premium: Investors do not have all the issuers information.

Over time, part of this excess return is destined to disappear, the Cat Bond market will become more ecient,with investors' improved familiarity with Cat Bond, its demand will increase, see e.g., Sterge and van derStichele (2016).

7We ignore various possible explanations for the decrease of the risk premium such as the increasingdemand and familiarity of investors with respect to exotic asset classes, see Bantwal and Kunreuther (2000),

4

in 2015 compared to an average of 8.3% over the previous 14 years.8 In summary, historicalreturns have been inated by lower than expected losses and future returns are expected todecrease due to the growing popularity of Cat Bonds.

Although diversication oered by Cat Bonds remains attractive to investors, these are lessappealing because diversication comes with a higher cost. Some authors advance the ideathat Cat Bonds have a zero correlation with traditional assets, e.g., Hoyt and McCullough(1999) among others. It is important to note that this diversication eect can not bereplicated by buying stocks or bonds of some insurance or reinsurance companies. Becausemost of these rms' assets are invested in traditional investments, they are highly correlatedwith the nancial markets. Constantin (2011), Clark et al. (2016), Carayannopoulos andPerez (2015) among others, arrive at similar conclusions about the correlation of Cat Bondswith the capital markets. Indeed, Cat Bonds have been proven not to be completely immuneto nancial turmoils. Similar to other assets (e.g., Chua et al. (2009), Page and Panariello(2018)), Cat Bond correlations with other investments are small in normal times, but incrises, they increase too, albeit to a lesser extent (Constantin, 2011). Using a DCC model,Clark et al. (2016) nd evidence that there is a dynamic correlation between Cat Bonds andthe markets. While rising during nancial crises, the correlations also increase following amajor disaster (e.g., Katrina) .9 Employing a multivariate GARCH model, Carayannopoulosand Perez (2015) nds that the zero correlation assumption holds in normal times, but not attimes of crisis. Nevertheless, Cat Bonds diversication eects show up when one needs them.In fact, their returns are less aected by the 2008 nancial crisis than those of traditionalassets.10

In sum, although Cat Bonds seem to oer returns commensurate with the risk incurred, they

Muermann (2008) among others.8Sterge and van der Stichele (2016) oer more explanations for the high returns compared to those

obtained from traditional assets. Since 2001, realized losses have been 40% of what were projected. Theexcess returns can therefore be explained in part by the fact that the Cat Bonds market has been sparedfrom major catastrophes. To substantiate this point, Sterge and van der Stichele (2016), by means of anex-ante simulation model, generate a serie of catastrophic events. From this simulation, they obtain a Sharperatio that goes from 2.4 to 0.5 and the volatility from 2.8 to 5.4%.

9Seasonality is another important feature of Cat Bonds. There appears to be a seasonality in Cat Bondsreturns during the hurricane season. This makes sense since nearly 60% of Cat Bonds issued are exposed tohurricane risk (Sterge and van der Stichele, 2016).

10Junk bonds exhibit a performance similar to the one from Cat Bonds most of the time, but in turmoils,they are much more aected. Also, Clark et al. (2016) show, in the context of a portfolio, that adding CatBonds provides diversication benets. Correlation results go against the idea that a shock in the marketdoes not aect the hurricane probability in Florida. This premise would have been true had Cat Bondsbeen immune from market and credit risks. However, it was not the case during the 2008 nancial crisis,(Krutov, 2010). In fact, the main cause of the increase in correlation is that assets held in the collateraltrusts were not as safe as expected and therefore, Cat Bonds were not fully immunized against credit risk,see e.g., (Carayannopoulos and Perez, 2015). This has led to changes in the structure of the Cat Bonds tofurther isolate them from the credit and market risks (Krutov, 2010). This new structure has been positivelyreceived by the market and the correlation appears to have returned to the pre-crisis level. The increase inthe correlation seems to have disappeared as of 2011 (Carayannopoulos and Perez, 2015). It remains to beseen whether this new structure could warrant a near zero correlation in the event of another crisis.

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remain attractive for the diversication benets in a portfolio context.

3 Methodology

We rst explain the dierent mean-variance (M-V) spanning tests. Then, we introduce themethodology used to quantify the variation of the diversication eects. Finally, we performstochastic dominance (SD) tests.

3.1 Mean-Variance Spanning Tests

The spanning tests introduced by Huberman and Kandel (1987) are generally considered asthe standard method for studying the diversication benets of nancial assets.11 It examineswhether the ecient (mean-variance) frontier obtained from the new assets combined with thebenchmark assets is spanned by the ecient one derived from those reference (benchmark)assets. In other words, we can say that the reference assets span the new assets combinedwith themselves if their ecient frontier is statistically identical to the one newly constructed.This test addresses the question whether an investor can benet by investing in this new setof assets. It uses a regression assuming K reference assets as well as N candidate assets tobe tested. In our case, N=1. In Huberman and Kandel (1987), the relationship between thereference assets and the tested assets is:

R2t = α + βR1t + εt (1)

with t=1, 2, . . . , T. In matricial form R1 is a T x K matrix of the reference assets' returns,R2, a T x N matrix of the test assets' returns, β, a K x N matrix the regression' coecients,α, a vector of size N and ε, a vector of size T of the error terms.

3.1.1 Mean-Variance Spanning Tests Under Normality

Following Huberman and Kandel (1987), we test the null hypothesis of whether the referenceassets span the new assets as follows:

H0 : α = 0N , δ = 1N − β1K = 0N . (2)

Equation 1 can be written as a multivariate linear regression:

Y = XB + E (3)

11For an early survey on testing for mean-variance spanning, see DeRoon and Nijman (2001). Further,Arvanitis et al. (2018) develop a test procedure for stochastic spanning for two nested portfolio sets based onsub-sampling and linear programming and apply this procedure to standard historical stock market returnsto reconrm an important role for higher-order moment risk in portfolio theory and asset pricing. As statedearlier, we will take on this issue by the stochastic dominance approach.

6

Y is the T x N matrix of R2t, X a T x (K+1) matrix with its rows [1, R′1t], B=[α, β]' and Ethe T x N matrix of εt. From this formulation, we have the classical OLS estimators:

B = (X ′X)−1(X ′Y ) (4)

Σ =1

T(Y −XB)′(Y −XB) (5)

A =

[1 0′K0 −1′K

](6)

C =

[0′N1′N

](7)

Θ = AB + C (8)

G =

[1 + µ′1V

−111 µ1 µ′1V

−111 1K

µ′1V−111 1K 1′K V

−111 1K

](9)

with µ1 = 1T

∑Tt=1R1t and V11 = 1

T

∑Tt=1(R1t − µ)(R1t − µ)′. Later, Kan and Zhou (2012)

dene:

G =

[1 + a1 b1b1 c1

]. (10)

For more details on the tests under the returns normality assumption, see Huberman andKandel (1987) and Kan and Zhou (2012). For the rest of this paper, we focus on tests withthe assumption of conditional heteroscedasticity.

3.1.2 Mean-Variance Spanning Tests Under Conditional Heteroskedasticity

Returns are assumed normally distributed for the tests described by Huberman and Kandel(1987). However, in many cases, assets returns exhibit heteroskedasticity. To tackle thisissue, Kan and Zhou (2012) develop a GMM test using the same regression framework.12

12There is a large literature on mean-variance tests. For instance, Chen et al. (2011) apply meanvariancespanning tests to nd that adding volatility-related assets does lead to a statistically signicant enlargementof the investment opportunity set for investors. Also, by means of the Kan and Zhou (2012) spanningtests, Belousova and Doreitner (2012) evaluate the diversication contribution of several commodities to aportfolio of traditional assets from the perspective of a euro investor. Similarly, we employ these spanningtests by virtue of their robustness in addressing our research question on the diversication benets of CatBonds.

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Kan and Zhou (2012) dene xt = [1, R′1t]′, εt = R2t − B′xt and gt = xt ⊗ εt. The dierent

moments are given by:

gT (B) =1

T

T∑t=1

xt ⊗ (R2t −B′xt). (11)

The GMM estimator of B is given by the minimization of gT (B)′S−1T gT (B). Newey and West(1986) dene an autocorrelation consistent estimator of ST :

ST = Ω0 +m∑j=1

kj(Ωj + Ω′j) (12)

where kj = 1− j/(m+ 1), m is dened as the lag and Ωj = 1T

∑Tt=j+1 gtgt−j. In our case, we

will use the estimator ST with an automatic lag selection from Newey and West (1994).

Since both estimates of B and Θ do not depend on ST , they remain the same as the OLSestimators in Equations 4 and 8. Kan and Zhou (2012) develop the GMM version of theWald test as follows:

Wa = Tvec(Θ′)′[(AT ⊗ IN)ST (A′T ⊗ IN)]−1vec(Θ′) ∼ χ22N (13)

where

AT =

[1 + a1 −µ1V

−111

b1 −1′K V−111

]. (14)

To improve the nite sample properties, we use the following F-test with N=1

Fa =

(T −K − 1

2

)Wa

T∼ F2,T−K−1. (15)

3.1.3 Step-DownMean-Variance Spanning Tests Under Conditional Heteroskedas-

ticity

Kan and Zhou (2012) show that 1- the tests' power is mainly driven by the dierence betweenthe global minimum-variance portfolios and 2- the distance between of the tangent portfolios(i.e., an increase in the slope of the tangent lines) is relatively unimportant. In other words,a small improvement in the standard deviation of the global minimum-variance portfolio maylead to rejecting the null hypothesis even if the economic dierence is not signicant. On theother hand, a big distance between the tangent portfolios will not bring about the rejection ofthe null hypothesis even if this dierence is economically signicant. The fact that statisticalsignicance does not always correspond to economic importance suggests using an approachwhere each component (α = 0N and δ = 0N) is tested separately.

8

Hence, following Kan and Zhou (2012), we rst test the hypothesis H0 : α = 0N thenH0 : δ = 0N given α = 0N . If the rst test is rejected, we conclude that the tangent portfoliosare statistically dierent. If the second test is rejected, then, the minimum variance portfoliosare statistically dierent. The GMM test is outlined in Appendix A.

3.2 Sub-Sample Analysis

Since asset characteristics vary across dierent regimes, investors adjust their portfolio allo-cations according to the market environment (i.e., dynamic asset allocation, see for example,Ang and Bekaert (2004) among others). To study the benets of adding Cat Bonds to a port-folio under dierent market cycles, we split the sample into dierent periods. To characterizechanges in regimes, we use three dierent methodologies. First, we identify the regimes usingthe Mahalanobis distance à la Kritzman and Li (2010). Second, we detect regimes employinga Markov-Switching model (Hardy, 2001). Finally, we depict US business cycles as denedby the National Bureau of Economic Research (NBER).

3.2.1 Mahalanobis Distance

To proxy nancial turbulence, Kritzman and Li (2010) use the Mahalanobis distance. Theydene nancial turbulence as episodes in which asset prices, in view of their historical behav-iors, behave in an unusual way. This measure incorporates both the volatility of individualassets and the correlation between them. Kritzman and Li (2010) dene the turbulence indexas:

dt = (yt − µ)Σ−1(yt − µ)′ (16)

where

dt = Turbulence during t (a scalar).

yt = Vector of asset returns for the period t.

µ = Vector of asset returns means.

Σ = Covariance matrix of asset returns.

To obtain the threshold separating observations during the turbulent market from those of thequiet market, a certain percentage (75% in our case) of the multivariate normal distributionmust be xed. "Calm" or "quiet" observations are therefore those bound by this percentageof the distribution. The observations outside this range are deemed "turbulent". Figure 1shows the turbulence index with a 75% threshold (represented by the horizontal line).

3.2.2 Markov-Switching

Regime-switching models are used to detect market regimes shifts between periods of highexpected returns with low volatility and those of low expected returns with high volatility.Asset returns distributions vary according to the corresponding market regimes, often by

9

means of a Markov chain. For simplicity, we use Hardy (2001) model, for a review of thislarge literature, see e.g., Kim and Nelson (1999), Hamilton (2010), Ang and Timmermann(2012).

The model assumes that time elapses discretely: t=0, 1, . . . , T. The regimes are modeled bya Markov chain with ρt representing the regime between t and t+1. There are K dierentstates of the market (K=2 in our case) so ρt can take values ranging from 1, 2, . . . , K. TheMarkov chain transition matrix P is:

Pi,j = P [ρt+1 = j|ρt = i], i, j ∈ (1, ..., K). (17)

Asset returns are dened as:

rt+1 = ln

(St+1

St

)= µρt + σρtεt+1 (18)

where St is the price at t, µj and σj are the mean and volatility parameters associated withregime j (j=1, . . . , K). ε is a sequence of i.i.d normal (0,1) variables. ε and ρ are independent.The distribution rt+1, conditionally on ρt = k, is normal with mean µk and standard deviationσk.

rt+1|ρt = k ∼ N(µk, σ2k). (19)

As in Hardy (2001), the maximum likelihood method is used to estimate the parameters.Once the parameters determined, we obtain the most probable sequence of regimes using theViterbi (1967) algorithm. For our purpose, we apply a regime-switching model to the returnsfrom the S&P 500 index and the Bloomberg Barclays US Aggregate Bond index. Figure 2presents the most probable regimes sequences of these indexes.

3.2.3 Business Cycles

We also perform our analyzes based on various business cycles that have been called by theNBER, http://www.nber.org/cycles/cyclesmain.html). For our period of study, there isonly one recession lasting from December 2007 to May 2009. Figure 3 displays these dierentregimes (1=recession).

3.3 Time Varying Diversication Benets

Changes in the correlations of Cat Bonds with other assets and risk factors suggest thebenets of diversication is variable. We describe next the methodology used to quantify thevariations in the diversication benets associated with adding Cat Bonds to a portfolio ofassets and factors.

3.3.1 Dynamic Conditional Correlations (DCC)

As shown in Section 4.4, correlations between Cat Bonds and other assets vary over time.Hence, as Clark et al. (2016), we use the dynamic conditional correlation (DCC) model

10

introduced by Engle (2002).13

With asset volatilities depicted by the Generalized Autoregressive Conditional Heteroskedas-ticity (GARCH) model (Bollerslev, 1986), the correlation matrix follows a GARCH-like pro-cess. Based on a two-step estimation, the DCC model is computationally more ecientthan a multivariate GARCH model. In addition, the DCC model presented in details in Ap-pendix B oers advantages over other correlation modeling techniques such as rolling-windowcorrelations because of its asymptotic properties (Engle and Sheppard, 2001).

3.3.2 Variations in the Sharpe Ratio Increases

To gauge the variation of the benets associated with the addition of Cat Bonds to a portfolioof assets, we use the technique proposed by Huang and Zhong (2013). It is based on theincrease in the Sharpe ratio from the tangent portfolio. The tangent portfolio is expressedas:

maxω

µ′ω − rf√ω′Σω

(20)

where ω are the portfolio weights, µ the return means, rf is the risk-free rate, Σ the covariancematrix, and ' the transpose operator.

Like Huang and Zhong (2013), to quantify the diversication benets, we use the dierencebetween the Sharpe ratios of the portfolio with Cat Bonds and the one without these. Wefollow their methodology:

1. Estimate the mean of the returns individually.

2. Estimate the DCC model.

3. Calculate the Sharpe ratio on a weekly basis using Equation 20 to obtain the portfolioweights with µ=step 1 and Σ=step 2. rf is represented by the 1 month US T-Bill rate.

4. Do steps 1 to 3 for a portfolio with and without Cat Bonds.

5. Calculate the dierence between the Sharpe ratios.

6. Repeat steps 1 through 5 for the remaining points in time to obtain a time series ofthe increases in the Sharpe ratio.

These steps will be executed for an asset-based allocation as well as for a factor-based allo-cation. In addition, to reect typical investors constraints, we perform these for a portfoliowhere short sales are not allowed.

13The asymmetric DCC (ADCC) model (Cappiello et al., 2006) is also considered, but the asymmetryparameter is not signicant.

11

3.3.3 Variations in the Higher Diversication Ratios

To overcome some drawbacks with the mean-variance framework14, following Choueifaty andCoignard (2008), we adopt the maximum diversication portfolio paradigm. The diversica-tion ratio is dened as:

DR =ω′V√ω′Σω

(21)

where V=

σ1...σn

is the volatility matrix.

The diversication ratio is therefore the weighted average of the individual volatilities dividedby the volatility of the portfolio. The maximum diversication can be written as:

maxω

DR. (22)

To quantify the diversication benet of adding Cat Bonds to a portfolio, we take the fol-lowing steps:

1. Estimate the DCC model.

2. Calculate the diversication ratio on a weekly basis using Equation 22 to obtain theportfolio weights with Σ=Step 1 and V are the individual volatilities in the model.

3. Do Steps 1 and 2 for a portfolio with and without Cat Bonds.

4. Calculate the dierence between the squared diversication ratios.

5. Repeat Steps 1 to 4 for the remaining instants to obtain a time series of the squareddiversication ratio's increases.

We carry out these steps for both the asset-based and factor-based allocations. When shortsales are allowed, the optimization tends to allocate too much weights to assets for whichvolatility has been underestimated, e.g., Michaud and Michaud (2008). To alleviate thisproblem, we also allow for a restriction on short selling, a potential investors' constraint. Wepresent only the increases in the squared diversication ratio results. Choueifaty et al. (2013)interpret the squared diversication ratio (DR2) as the number of independent risk factorsin the portfolio.

3.4 Stochastic Dominance Tests

To expand beyond the classic mean-variance framework for comparing risk and performance,we employ a stochastic dominance eciency (SDE) approach. The SDE concept is an exten-sion of the stochastic dominance (SD) one. SD is a criterion used to compare two portfolios,

14For example, while the variance can generally be estimated with a fairly high level of condence, theaverage returns (µ) are much more dicult to estimate accurately.

12

see for example, Levy (2015), Sriboonchita et al. (2009), Post (2016) among numerous others.Being non-parametric, SD does not require any assumptions about the returns distribution.Moreover, unlike the mean-variance context, it does not impose explicit specication on in-vestors' preferences. The rst-order stochastic dominance (FSD) and second-order stochasticdominance (SSD) are dened as:

FA(x) ≤ FB(x) (23)∫ ∞−∞

[FA(x)− FB(x)]dx ≤ 0 (24)

where F is the probability cumulative function of returns and a portfolio A has stochasticdominance over a portfolio B. A stochastic dominant portfolio under the rst-order criterionconstitutes a necessary condition for the second-order, but the opposite is not true (Levy,2015). Figure 4 shows an example of FSD and SSD. In the next section, we describe the SDEtests and show how to implement these.

3.4.1 Tests Description of the Stochastic Dominance Tests

We employ the stochastic dominance eciency test introduced by Scaillet and Topaloglou(2010).15 The returns of the n assets are described by the stationary process Yt takingvalues in Rn with t=1, . . . , T. Let F(y) be the cumulative function of Y=(Y1, . . . , Yn)' atpoint y=(y1, . . . , yn)'. Consider a portfolio of n assets and the vector λ of its weights withλ ∈ L where L= [λ ∈ Rn :

∑λ = 1]. Denote G(z,λ;F) as the probability cumulative function

of portfolio λ′Yt at point z with:

G(z, λ, F ) =

∫Rn

I[λ′u ≤ z]dF (u) (25)

where I is an identity function taking the value 1 if λ′u ≤ z and 0 otherwise. Let us dene:

J1(z, λ, F ) = G(z, λ, F ) (26)

J2(z, λ, F ) =

∫ z

−∞G(u, λ, F )du =

∫ z

−∞J1(u, λ, F )du (27)

and so

Ji(z, λ, F ) =

∫ z

−∞Ji−1(u, λ, F )du. (28)

15There is a large literature on stochastic dominance. However, we use this type of test because it allowsus to determine whether a portfolio dominates an innite number of combinations of others. In addition, itis more suitable for our Cat Bond world of fat-tailed and autocorrelated returns.

13

The assumptions for the SDE test at orders j=1 (rst-order) and j=2 (second-order) can bewritten as:

Hj0 : Jj(z, τ, F ) ≤ Ji(z, λ, F ) for all z ∈ R and for allλ ∈ L (29)

Hj1 : Jj(z, τ, F ) > Ji(z, λ, F ) for some z ∈ R and for someλ ∈ L. (30)

Under the null hypothesis, no λ portfolio dominates the benchmark τ at order j (τ is saidto be SDE). Jj(z, τ, F ) is always less than or equal to the function Jj(z, λ, F ) regardless of λportfolio and point z. We test the null hypothesis at order j using the Kolmogorov-Smirnovtest (Scaillet and Topaloglou, 2010):

Sj =√Tsup

z,λ[Jj(z, τ, F )− Jj(z, λ, F )] (31)

where F is the empirical distribution of F:

Jj(z, λ, F ) =1

T

T∑t=1

1

(j − 1)!(z − λ′Yt)

j−1I[λ′Yt ≤ z]. (32)

We can therefore reject Hj0 if Sj > cj, with cj being the critical value. To calculate the

p-value associated with the critical value cj, we follow the steps described in Section 3.4.3.The properties of this test can be found in Scaillet and Topaloglou (2010).

3.4.2 Implementation

First-Order Stochastic Dominance

The rst-order stochastic dominance (FSD) null hypothesis amounts to E[U(τ)] ≥ E[U(λ)]for any non-decreasing utility function U(z) of z (U ′(z) > 0). The rst-order stochasticdominance eciency test (FSDE) S1 can be solved by mixed integer programming (MIP).Its full formulation is below:16

maxz,λ

S1 =√T

1

T

T∑t=1

(Lt −Wt)

s.t. M(Lt − 1) ≤ z − τ ′Yt ≤MLt, ∀t ∈ TM(Wt − 1) ≤ z − λ′Yt ≤MWt, ∀t ∈ T∑

λ = 1

Wt ∈ 0, 1, Lt ∈ 0, 1, ∀t ∈ T

(33)

16Using a procedure to compute p-values described later in 3.4.3 and solving this, all together take an hourlong, which makes the FSDE computationally prohibitive.

14

with M being a large constant. 1T

∑Tt=1 Lt and

1T

∑Tt=1Wt represent J1(z, τ, F ) and J1(z, λ, F )

respectively. This formulation enables us to test the dominance of a given τ portfolio on anyλ portfolio combination. For details, see Appendix C.

Second-Order Stochastic Dominance

The second-order stochastic dominance (SSD) null hypothesis consists of E[U(τ)] ≥ E[U(λ)]for any non-decreasing monotonic and concave utility function of z, i.e., U ′(z) > 0 andU ′′(z) < 0. The second-order stochastic dominance eciency test (SSDE) S2 can also besolved by MIP (mixed-integer programming) in less a minute, see Appendix D for the com-plete formulation.17

3.4.3 P-Value Simulation

To obtain the p-value associated with the critical value cj, we use the bootstrap simulationmethod proposed by Scaillet and Topaloglou (2010). To create the dierent samples needed,we employ the nonoverlapping block bootstrap technique from Carlstein (1986). The advan-tage is that it does not assume the returns to be i.i.d.. Let b and l be integers such that T=b*l.Then, split the data into b distinct blocks, the k th block being Bk = (Y(k−1)∗l+1, . . . ,Yk∗l)with k ∈ 1, 2, . . . , b. We then choose b blocks randomly with replacement. The sampleYt∗ is therefore composed of b blocks B1

∗, ...,Bb∗ selected randomly with replacement. Yt

∗

is our block bootstrap sample. We dene l=T 1/3 and generate R=300 bootstrap samples.We can estimate the p-value by:

pj =1

R

R∑r=1

I[Srj > Sj] (34)

where Srj is the statistic of the test obtained using the r th bootstrap sample.

3.4.4 Procedure

To determine whether a portfolio of asset classes (or factors) stochastically dominates aportfolio within the same universe with the addition of Cat Bonds, we use the two-stepprocedure proposed by Daskalaki et al. (2017).

In the rst step, we dene the US Equity (USEQ) index returns as the τ portfolio for theasset-based allocation (Equity market for the factor-based). Next, we test whether the τportfolio stochastically dominates all λ portfolios of asset classes (or factors).

In the second step, we use the portfolio obtained in the rst step as the new benchmark τ ,i.e., the USEQ portfolio (or Equity market for factors) if the null hypothesis is not rejected

17A MIP problem is one where some of the decision variables are constrained to be integer values at theoptimal solution.

15

in Step 1 or the λ portfolio that maximizes Jj(z, τ, F ) − Jj(z, λ, F ) if the null hypothesisis rejected. Finally, we perform the stochastic dominance eciency tests for τ against allpossible λ combinations of the universe of assets classes (factors) plus Cat Bonds.

These steps require solving 2*(R+1) programming problems. This task would require about2*(R+1) hours because to solve a single FSDE problem takes an hour. Therefore, we willonly perform the SSDE tests. It will be applied for an allocation based both on asset classesand factors. Also, it will be tested on the entire sample as well as on dierent sub-samplesdened in Section 3.2.

3.5 Out-of-Sample Analyses

So far, all the tests were conducted in-sample. Now, we check whether the results can betransposed out-of-sample by means of a rolling window technique of length K. Assume thatwe have data series of T observations. We use data 1 to K to nd the portfolio weights forthe period [K, K + 1]. From these weights, we can calculate the portfolio's realized returnusing the observed returns for that period. Then, we use the following K data by droppingthe rst data (2 to K + 1) to obtain the weights for the period [K + 1, K + 2]. Using theobserved returns for this period and our weights, we get our portfolio return for [K + 1, K +2]. We proceed in this manner up to T. We obtain a series of (T-K) returns for our portfolioto evaluate out-of-sample performance. For the analysis, we use K = 120.

The out-of-sample analysis will be made for portfolios of asset classes with and without CatBonds and also for portfolios of factors with and without Cat Bonds. We calculate theweights for each of these assets (factors) using three dierent methodologies: the maximumSharpe ratio portfolio, the maximum diversication portfolio, and the second-order stochasticdominance portfolio.

We simply construct the maximum Sharpe ratio portfolio by nding the (1st and 2nd) mo-ments of the returns from our given K interval data and using Equation 20. We obtain themaximum diversication portfolio in a similar fashion except that we use this time Equation22 and the volatility and correlation matrix of the returns on the interval. For the con-struction of the second-order stochastic dominance portfolio, we apply the 2-step procedureemployed by Daskalaki et al. (2017) for the SSDE criterion 3.4.2 on each of our sample datasubset of length K (computed with the cumulative probability function of returns).

3.5.1 Performance Evaluation

Next, we compare the performance of the portfolios with and without Cat Bonds usingdierent performance measures: the Sharpe ratio (SR), the portfolio turnover, the conditionalSharpe ratio (CSR) and the Omega ratio.18 These metrics enable us to assess benets ofadding Cat Bonds from an out-of-sample perspective.

18For other performance measures based on partial moments of the distribution, see Zakamouline (2014),Kaplan and Knowles (2004) for instance.

16

Sharpe Ratio

The Sharpe ratio (see for instance, Lo (2002) among numerous others) is the average of theexcess returns of our portfolio (µi − rf ) divided by the standard deviation of these returns(σi).

SRi =µi − rfσi

. (35)

To test whether the Sharpe ratios of two portfolios are statistically dierent, we use the testfrom Memmel (2003). The null hypothesis is the Sharpe ratios are equal. Using only theSharpe ratio as a performance criterion can be misleading because it only accounts for therst two moments and ignore the higher ones that are often signicant as buttressed by ourdescriptive statistics.

Portfolio Turnover

To approximate the amount of trading needed to implement the strategies, following DeMiguelet al. (2007), we use the portfolio turnover:

PT =1

T −K

T−K∑t=1

N∑i=1

|ωi,t+1 − ωi,t+| (36)

where N is the number of assets in the portfolio, T-K is the number of periods were theportfolio allocation is changed, ωi,t+1 is the weight of the asset at t+1 and ωi,t+ is the weightof the asset at the end of period t just before re-balancing the portfolio allocation to takeinto account the return obtained during period t to t+1. It is calculated as:

ωi,t+ =ωi,t(1 + ri,t)

1 +∑N

i=1 ωi,tri,t. (37)

A high turnover entails signicant transaction costs which lower realized returns.

Conditional Sharpe Ratio

The conditional Sharpe ratio is the mean of our portfolio excess returns (µi− rf ) divided bythe conditional value-at-risk (CVaR) at an α level. The CVaR is dened as the average oflosses knowing that a certain threshold (VaR) has been exceeded.19 It is calculated using theCornish-Fischer expansion to account for the returns asymmetry and kurtosis, e.g., (Maillard,

19The use of the CVaR risk measure allows to take into account the information contained in the tails ofthe distribution which are ignored by the ordinary Sharpe ratio.

17

2018).

CSRi(α) =µi − rf

CV aRi(α). (38)

Omega Ratio

The Omega ratio introduced by Keating and Shadwick (2002) is another performance mea-sure widely used (see for instance, Kazemi et al. (2004)) that overcomes the fundamentaldeciency of the Sharpe ratio, which considers that returns are normally distributed. Theadvantage of this ratio is that it considers the entire distribution of the portfolio returnswithout making any parametric assumptions (see Kaplan and Knowles (2004), Zakamouline(2014) for generalisations of the Omega ratio). It is calculated by splitting the cumulativeprobability function into two partitions to depict a loss region and a prot region with respectto a threshold as follows:

Ω(L) =

∫ bL(1− F (x))∂x∫ LaF (x)∂x

(39)

where L is the threshold demarking the region of losses in relation to the one of gains andF() is the cumulative probability function of the portfolio returns dened over the interval[a, b]. For a given L level, an investor should prefer the portfolio with the highest Omegavalue. Comparing two portfolios should not be made at a single threshold but over a rangeof L. Therefore, we perform our analyses by means of Omega curves. These curves intersectat the so called points of indierence. The slope of the function is usually an indication ofthe risk. The higher the inclination, the less risky is the portfolio. However, above its mean,a high slope entails a low earnings potential. L can also represent a level of risk aversion.The lower it is, the higher the risk tolerance. Lacking a reference threshold L, it is necessaryto compare the Omega values of two portfolios by considering all these features.

4 Data

In this section, we describe the dierent datasets used to represent the Cat Bonds and thedierent assets composing the portfolios. We compile our data from various standard sources,including Bloomberg, Datastream, Federal Reserve Economic Database and the data librarieson Ken French's website. We use weekly data from January 4, 2002 to June 30, 2017. Returnsare calculated as follows:

Rt = ln

(St+1

St

).

18

4.1 Cat Bonds

We use indexes compiled by Swiss Re to represent Cat Bonds. The Swiss Re Cat Bond Perfor-mance Indices are designed to reect the returns of Cat Bond markets.20 Their constructionmethodology is described in Swiss Re (2014). They are the only such indices provided ona weekly basis. We consider ve baskets tracking total returns, which are composed of thecoupon return (accrued stated spread plus collateral return) and the price return (measuringthe movement of secondary bid indications) Panel A of Table 1 presents the list of theseindexes and their respective descriptions. Data for some Cat Bonds/ILS funds are available.However, the frequency of these data is monthly or only available for a short time length. Tocarry out a robust analysis which is only possible with more data, we resort to the indexesproduced by Swiss Re.21

4.2 Asset Class and Factor Portfolios

Recall that our objective is to test the eects of adding Cat Bonds to the assets available toinvestors. Our rst reference portfolio imitates an allocation based on asset classes. Panel Bin Table 1 lists the assets and indexes used to compose the portfolio.

Following the 2007 nancial crisis, allocation by asset classes has become less popular. Infact, during the crisis, this type of allocation failed to provide diversication when needed.Asset classes, previously with low correlations, became much more correlated, thus losing, ina greater extent, their diversication benets in turbulent times. This is why a factor-basedallocation has become preferable, advocated and promoted, see for instance, Ang (2014), Naiket al. (2016). It aims to identify persistent attributes and risk-return dimensions of nancialassets that best explain their returns instead of selecting assets belonging to separate classes[Bender et al. (2010), Page and Taborsky (2011), Clarke et al. (2016), and Blanc-Brude et al.(2017)].

Consequently, we perform our analysis under both an asset-based allocation context and afactor-based one constituting our second reference portfolio. Panel C of Table 1 presents thelist of the dierent factors and indexes used for the portfolio construction.

4.3 Descriptive Statistics

Tables 2 and 3 present the descriptive statistics of the various indexes for the full sample andduring the nancial crisis (12/2007-05/2009) respectively.

Average weekly returns for the Cat Bonds indexes vary between 0.12% and 0.16%. Theirvolatility is around 0.36% except for SRUSWIND which is 0.72%, therefore, it gives a Sharpe

20We assume that bonds contractual, geographical, market liquidity and credit cycles factors to be sub-sumed in these returns. Note further that life and health Cat Bonds, including excess mortality, excesslongevity and morbidity-linked bonds, are not contained within these indices.

21As mentioned earlier, these indexes are not investible, however, Jaeger et al. (2011) provide a methodologyfor creating an investible index and Trottier et al. (2019) present a characterization of Cat Bond performanceindices.

19

ratio lower than the other Cat Bonds indexes (0.192 vs [0.254 0.368]). This may suggest thatthe impact of Cat Bonds depends on the type of perils they cover (see Braun et al. (2019)).Some assets or factors provide average returns similar or greater than those from Cat Bonds.However, they generally have higher volatility and therefore have a smaller Sharpe ratio. Theratios most similar to the ones from Cat Bonds are oered by USCB, USHY and C.CURVE(0.116, 0.124 and 0.146). In a mean-variance context, Cat Bonds could oer good potentialreturns. In addition, they had a maximum of 19% negative returns. The next asset with theleast number of negative returns is USHY with 33%, which is substantial compared to thoseof Cat Bonds. Remarkably, Cat Bonds display a very high kurtosis involving fatter tails thannormal, which could have an impact when the whole distribution is considered instead ofonly the mean and the standard deviation.

When we look at these same statistics during the last nancial crisis, we note that Cat Bonds,unlike most assets, have maintained positive average returns and a low number of negativereturns. They also maintained a low volatility and therefore a high Sharpe ratio relative toother assets. This seems to suggest a potent diversication contribution in times of crisis.However, the average return decreased signicantly ([0.03% 0.07%] vs [0.12% 0.16%]), whichproves that Cat Bonds returns were not immune during the crisis. Only two asset classesout of eight have a positive mean (USCB very close to 0). These classes still had 46% and49% negative returns. On the other hand, only ve out of ten factors had negative meanswith only three having more than 50% negative returns. This suggests that a factor-basedallocation is resiliently more robust to nancial turmoils.

In summary, the risk-return relationship gives the impression that Cat Bonds are attractiveinvestment opportunities. Nonetheless, the very high kurtosis could exert an impact on theirinvestment performance when the whole distribution is considered rather than only the rsttwo moments.

4.4 Correlations

Table 4 presents the correlation matrix between Cat Bonds indexes and various asset classesreturns for the full sample as well as the one between Cat Bonds and the chosen factorsreturns for the same period of study. First, we notice that the correlation between thedierent Cat Bonds indexes returns is very high. Yet, for the SRUSWIND index, it is lower(between 73.1% and 80%). The average correlation between the SRCAT index and the assetclasses is 9.79%, which seems to support the hypothesis that Cat Bonds provide diversicationbenets. For the SRUSWIND index, the average correlation with the other asset classes is5.38%. Second, the average correlation between the factors and SRCAT and SRUSWIND is2.88% and 2.01% respectively. The average correlation between the dierent asset classes is28.06% versus 0.01% for the factors. This is in agreement with the existing literature whichsuggests the advantage of a factor-based allocation.

Table 5 shows the matrices in question during the nancial crisis (12/2007-05/2009). Theaverage correlations of the SRCAT and SRUSWIND indexes relative to the asset classes are29.77% and 31.73%, a signicant increase from those obtained using the full sample. Thisagain points to the fact that Cat Bonds were not spared by the 2008 crisis. These correlations

20

with the factors reaching 5.84% and 7.28%, are smaller than for the asset classes. The averagecorrelation between asset classes is 33.79% and 0.07% between the factors. The associationbetween the dierent factors is more robust than the one between the considered asset classesduring the nancial crisis. These results imply that the correlations between the dierentassets and factors are time-varying lending support to our approaches that take into accountdynamic correlations.

5 Results

5.1 Mean-Variance Spanning

Table 6 provides the spanning tests results for the full sample for dierent Cat Bonds in-dexes. For all indexes and tests, spanning can be rejected at a level above 99% regardless ofwhether the portfolio contains asset classes or factors. This suggests that the ecient fron-tier is signicantly improved by adding Cat Bonds. Furthermore, results from the step-downtest show that Cat Bonds signicantly improve the tangent portfolio (SD1) as well as theminimum-variance portfolio (SD2).

Spanning tests for periods detected by a Markov-Switching model (Table 7) show that addingCat Bonds improves the mean-variance frontier, the tangent portfolio and the minimum-variance portfolio in episodes of high volatility. This is robust for a benchmark portfolioof asset classes and factors. However, we cannot reject spanning for the minimum-varianceportfolio with SRUSWIND index during the low volatility periods of the US aggregate bondindex. We conclude that in a mean-variance world, adding Cat Bonds to portfolios of assetsis benecial.

Tests results for periods determined by the NBER and by the turbulence index from Table 8show that, in general, spanning can be rejected regardless of the period, the index or portfoliobased universe. However, tests for the tangent portfolio cannot be rejected during the recentrecession for the SRUSWIND index under an asset class context and are only rejected ata level of 90% for the SRBBCAT index for an asset class portfolio and SRUSWIND withfactors during the recession. Moreover, in turbulent times, spanning of the tangent portfoliois not rejected for the SRUSWIND index for both of our asset and factor based portfolios.The under performance of the SRUSWIND index during recessions and turbulent periodshighlights the fact that diversication by type of peril (hurricanes, earthquakes, etc.) wheninvesting in Cat Bonds is important and that focusing on a single risk, here hurricanes in theUnited-States, may decrease the benets of adding Cat Bonds.

5.2 Time-Varying Diversication Benets

In this section, we analyze results of the time varying diversication benets of adding CatBonds to a portfolio.

21

5.2.1 DCC Correlations

As the correlation between the dierent Cat Bonds indexes is very high, Figures 5 and 6present only the correlations obtained from the DCC model of the asset classes and factorswith the SRCAT index respectively. Notice that the correlations vary considerably over time.Indeed, for asset classes, the correlation varies between -0.15 and 0.25 and for factors between-0.3 and 0.3. It is important to note that correlations for most asset classes peaked aroundthe 2008 crisis. This is in line with the literature documenting the signicant increase ofthe correlation of assets with Cat Bonds around the crisis. Following 2008, the correlationsseem to return to a low level. However, the correlation between Cat Bonds and US TreasuryBonds (USTB) shows an upward trend. We do not observe an increase in correlations aroundthe crisis of Cat Bonds with the factors. Indeed, although these vary over time, they do notseem to have a common trend. This underlines the advantage of factor investing which, inturbulent times, remains more stable than an asset-based allocation.

5.2.2 Variations in the Sharpe Ratios

Figure 7 shows the variation of the Sharpe ratio under an asset classes and factors context withand without allowing short selling. We can see that all Cat Bonds show a similar variationover time. In addition, we note a huge decline in all diversication benets, manifested hereby increases in the Sharpe ratio, around 2006 and 2007. This corresponds to the increaseof Guy Carpenter U.S. Property Rate on Line index following Hurricane Katrina shown inFigure 8.

Despite the increase in correlation around 2008 stark in Figure 5, the diversication benetsprovided by Cat Bonds remain substantial in these years.

Table 9 reports summary statistics of the Sharpe ratio increases. For all indexes and dierentportfolio types, Sharpe ratio increases are signicantly bigger than zero. In all cases, theSRUSWIND index was the one with the lowest average increase. Prohibiting short sellingdoes not change our results.

5.2.3 Variations in the Diversication Ratios

Another way to quantify the variation of the diversication benets is to look at the increasein the squared diversication ratio of the maximum diversication portfolio. Figure 9 showsthese diversication uctuations for the benchmark portfolios of asset classes and factors withand without short selling. The squared diversication ratio can be viewed as the portfolionumber of independent risk factors. An increase in this ratio would indicate whether addingCat Bonds brings a new risk factor to the portfolio. We also nd that, Cat Bonds, if combinedwith an asset class portfolio, appear to lose its diversication virtue around the 2008 crisis.This coincides with the increase in correlations between Cat Bonds and the asset classesconsidered. This is not the case with the factors where the increase seems to be morerandom. In general, we can conclude that Cat Bonds bring a new risk factor to a portfolioexcept during a crisis for a portfolio of asset classes where the dierence in the squareddiversication ratios fell below 0.5 for all indexes.

22

We display these increases summary statistics in Table 10. We note that the asset classportfolios average of the dierences in the diversication (squared) ratio is around one exceptfor the SRGLUCAT index. This means that the diversication benets are lower for thisindex. However, in all cases, the increase is signicantly larger than zero. Also, for a factor-based portfolio, the average increase is lower. This suggests that a factor-based allocationprovides greater diversication than an asset-based one, which is in line with the existingliterature.

5.3 Second-Order Stochastic Dominance Eciency (SSDE)

The null hypothesis for the second-order stochastic dominance eciency (SSDE) test is thata benchmark portfolio, here an asset class (or factor)-based portfolio, is ecient in termsof stochastic dominance over the same portfolio to which we add Cat Bonds. Panel A ofTable 11 exhibits results for the full sample, the NBER business cycles and the periodsdetermined by the turbulence index. Panel B shows the results for the periods estimatedby a Markov-Switching model on the S&P500 and the US Aggregate Bond Index. The nullhypothesis is not rejected for all periods and Cat Bonds indexes. It is surprising that mostof the spanning tests were rejected. These results show that to compare portfolios one has totake into account the whole distribution. The high kurtosis of the Cat Bonds indexes likelyplays a role in these tests results. In summary, when the entire distribution of returns isconsidered, adding Cat Bonds to a traditional portfolio of asset classes or factors does notlead to a new portfolio capable of dominating the old one under the second-order stochasticdominance criterion.

5.4 Out-of-Sample Results

In the previous sections, we show that in-sample Cat Bonds provide diversication benetsin a mean-variance context. However, when we consider the whole return distribution, weget the opposite conclusion. In this section, we discuss the portfolios out-of-sample perfor-mance results with and without Cat Bonds constructed by maximizing the Sharpe ratio, thediversication ratio or by using the second-order stochastic dominance.

Table 12 presents the Sharpe ratio, its p-value when the null hypothesis is that the ratiosare equal, the turnover as well as the conditional Sharpe ratio of the maximum Sharpe ratioportfolios with and without short selling built from asset classes or factors. Results for theSharpe ratio seem mixed. Indeed, for asset classes portfolios with short selling, the dierencefor portfolio with and without Cat Bonds is not signicant for two indexes. However, withoutshort selling, the hypothesis that the ratios are equal can be rejected for all indexes at a levelof 95% (only 3 at a 99% level). For factor-based portfolios, the Sharpe ratio results indicate abetter performance for the portfolios that include Cat Bonds. Indeed, the null hypothesis canbe rejected in all cases at a 90% level and 9 times out of 10 at a 99% level. The turnover of thebenchmark portfolios is always higher than the one of those that include Cat Bonds, whichcould increase the transaction costs incurred by benchmark portfolios. Performance underthe conditional Sharpe ratio suggests that investors does not always benet from adding Cat

23

Bonds in their investment portfolios. For example, for asset classes with short selling, none ofthe portfolios ratios with Cat Bonds exceeds their benchmark. However, in the case of assetclasses without short selling and both cases of factors, performance with Cat Bonds is betterfor four indexes in all situations. We also observe that the performance of the SRUSWINDindex is signicantly lower than the one of other indexes, which again supports the conclusionwe make in relation to the importance of a diversication by Cat Bonds hazard types.

Figure 10 provides the Omega ratio curves for the maximum Sharpe ratios portfolios with andwithout short selling built from asset classes or factors. Portfolios of asset classes with shortselling (top left) results do not show a signicant advantage over those from the portfolioswith Cat Bonds. We note a small advantage for very negative thresholds of four indexes,but a slight disadvantage for higher thresholds for all Cat Bonds indexes. The SRUSWINDindex under-performs its benchmark for almost everywhere. In the case where short sellingis prohibited (top right), all indexes except SRUSWIND show a signicant advantage inthe negative part, which suggests a better down side protection. For the case with factorsand without restrictions (bottom left), the SRUSWIND index once again under performs itsbenchmark for most of the thresholds. For the other indexes, we notice the same previouspattern, i.e., an advantage for the portfolios with Cat Bonds in the negative part and asignicant advantage of the benchmark in the positive part (in others words, the slope ishigher for those with Cat Bonds). This suggests that portfolios with Cat Bonds are generallyless risky. We draw the same conclusions for the case with factors under a short sellingconstraint (bottom right). In summary, the Omega ratio curves show that the benets ofadding Cat Bonds to the investing universe depend on the investors thresholds.

Table 13 shows the same results for the maximum diversication portfolios. Portfolios withCat Bonds yield signicant higher Sharpe ratios (at 95% level) in all contexts and higherconditional Sharpe ratios in 19 out of 20 times. However, the turnover from portfolios withCat Bonds are mostly higher than the one from their benchmark portfolio (17 of 20 times).

Figure 11 shows the graphs of the Omega ratios for these same portfolios. In the caseof portfolios constructed from asset classes (top left), we note that portfolios containingCat Bonds dominate signicantly in the negative part but slightly under perform for highthresholds, suggesting that they are less risky. However, the portfolio which includes the CatBonds SRUSWIND index underperforms for the vast majority of thresholds. We can drawthe same conclusions for the case without short selling (top right). For both our factors cases(bottom left and right), the portfolios with Cat Bonds Omega ratio does not seem to besignicantly dierent from their benchmark regardless of the threshold. These results meanthat, in certain situations, adding Cat Bonds is not really benecial for investors.22

Table 14 provides the performance results for portfolios built under the SSDE criterion.First, for portfolios of asset classes without shorting restrictions, the null hypothesis that theSharpe ratio with Cat Bonds equals the one of its benchmark portfolio can not be rejected

22Increased accessibility to investments in Cat Bonds, creative launching of ETFs, expanding availabilityof historical data of various specialized funds or direct investment data will enable future research to providemore conclusive evidence on the actual returns obtained by investing in these instruments.

24

at a level above 95% and can be rejected only 2 times at a level of 90%. Despite the fact thatfour times out of ve, this ratio is higher for portfolios with Cat Bonds, their turnovers areextremely high and always higher than those of their benchmarks. This implies that highertransactions costs could inuence the net performance of the strategy. Also, the fact thatthe benchmark portfolio's conditional Sharpe ratios are higher than those with Cat Bondsincluded implies that the high kurtosis of Cat Bonds returns brings about a higher tail risk.For the case without short selling, we nd that Cat Bonds have a positive impact on theportfolio (the null hypothesis of equal Sharpe ratios is rejected for all indexes except for theSRUSWIND index). Also, the turnover in these portfolios is much lower. However, it isimportant to note that the conditional Sharpe ratio of two indexes is below their benchmark.This again does not dismiss that the tail risk associated with Cat Bonds is not negligible. Forthe factor portfolios with short selling, Sharpe ratios of portfolios that include Cat Bonds arehigher than their benchmark and the hypothesis of equal Sharpe ratios is always rejected at alevel of at least 90% (4 out of 5 at a 95% level or higher). However, once again, the turnoverof these portfolios is very high compared to the one of their benchmarks. Also, all conditionalSharpe ratios are higher than those of their benchmarks. The case with restriction on shortselling shows that adding Cat Bonds does provide benets. In fact, all Sharpe ratios andconditional Sharpe ratios are higher than those of the benchmarks (with p-value <5 %) andall turnover ratios are lower as well.

The Omega curves of these portfolios are shown in Figure 12. For all asset class portfolioswithout restrictions (top left), the benchmark signicantly outperforms portfolios that includeCat Bonds for negative thresholds and slightly underperforms for the other part. For the casewithout short selling (top right), the opposite is observed, that is, a superior performance ofportfolios with Cat Bonds for the lower part and a slight under performance for the otherthresholds. However, this does not happen for the SRUSWIND index where the curve lookssimilar to the case with short selling allowed. For factors with short selling (bottom left),curves are essentially similar to the ones from asset classes with short selling allowed. Finally,when we restrict short selling (bottom right), the benchmark portfolio has a higher Omegaratio than the one of all portfolios including Cat Bonds for the vast majority of thresholds.In sum, these results reinforce the need to better qualify the benets of Cat Bonds andunderscore the importance of per-hazard (peril) diversication within Cat Bonds.

6 Conclusion

The most appealing attribute of Cat Bonds arises from the diversication these bonds mightbring to an investment portfolio. By constructing portfolios with various Swiss Re Cat Bondsindices and standard asset classes and factors over the period of 2002 to 2017, we revisit thisissue by investigating in-depth whether Cat Bonds do in fact oer diversication benets forinvestors. If there is a Cat Bonds diversication benet, to the best of our knowledge, weare the rst to provide the most comprehensive study on how it varies over time and looklike under dierent market conditions.

Toward this end, we perform various mean-variance spanning tests (Kan and Zhou, 2012) overdierent time periods for a universe of both asset classes and factors. The results indicate

25

that adding Cat Bonds gives rise to an ecient frontier, a minimum variance portfolio and atangent portfolio previously unattainable for all (quiet and turbulent) market conditions.

We use the dynamic conditional correlation (DCC) model from Engle (2002) to show thatcorrelations between Cat Bonds, asset classes and factors are not constant over time. Indeed,consistent with the extant literature, while we nd that correlations between Cat Bonds andasset classes are very low, these increased during the 2008 crisis. However, this is not thecase for correlations between Cat Bonds and factors. Using the DCC estimates, à la (Huangand Zhong, 2013), we calculate variations in the Sharpe ratios for the tangent portfolioover time when we add Cat Bonds to it. During the 2008 nancial crisis, we nd that theSharpe ratio (SR) increase is signicant for portfolios of asset classes and factors with andwithout a short selling constraint. However, the SR increase was not signicant over theyears following Hurricane Katrina. To investigate further Cat Bonds diversication benetsover time, we reuse the DCC estimates to calculate changes in the increases of the squareddiversication ratio (SDR) of the (Choueifaty and Coignard, 2008) maximum diversicationportfolio. Although the increase is signicant over the period, we notice a signicant drop inthe SDR around 2008 for the asset classes portfolios. This means that adding Cat Bonds doesnot always bring about another risk reducing factor to an asset class-based portfolio. Thisis not the case for a factor-based portfolio for which we get an increase in the diversicationbenet that varies over time but remains signicant for the whole period.

To analyze the eects of adding Cat Bonds without relying on assumptions on the distribu-tion of asset or factor returns and investors preferences, we use the second-order dominancestochastic eciency (SSDE) test proposed by Scaillet and Topaloglou (2010). We employthe two-step SSDE procedure developed by Daskalaki et al. (2017) for a portfolio of assetclasses and a portfolio of factors under a variety of market conditions. Based on our second-order dominance stochastic criterion empirical results for the dierent periods tested, wecannot reject the hypothesis that portfolios that do not include Cat Bonds are ecient. Thiscontradicts the results obtained by the mean-variance spanning tests and underscores theimportance of taking into account the whole distribution of returns. One possible cause ofthe ineciency as a result of adding Cat Bonds is that they inject a lot of kurtosis whichthickens the tails of the distribution. This eect is dicult to detect in a mean-varianceframework.

Finally, to ultimately test whether Cat Bonds do in fact have a positive impact, we conductan out-of-sample analysis. We employ three dierent portfolio construction techniques. Weperform the analysis for the maximum Sharpe ratio portfolio, the maximum diversicationportfolio and the portfolio optimized using Daskalaki et al. (2017) two-step SSDE portfolio.We build all these portfolios using a benchmark universe consisting of asset classes or factorswith and without short selling allowed. Results are not clearly conclusive since these dependon the context, the Cat Bonds indexes, the portfolio construction techniques, and the per-formance metrics used. This suggests that an investor is not always better o by adding CatBonds to his portfolio in the quest for diversication returns.

26

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31

Appendix

A Step-Down Mean-Variance Spanning Tests Under Conditional

Heteroskedasticity

The GMM test H0 : α = 0N is given by:

Wa1 = T α′(

([1 + a1, −µ1V−111 ]⊗ IN)ST ([1 + a1, −µ1V

−111 ]′ ⊗ IN)

)−1α ∼ χ2

N

where ST is the autocorrelation consistent estimator presented above. To improve the nitesample properties, we use the following F-test:

Fa1 =

(T −K −N

N

)Wa1

T∼ FN,T−K−N .

To perform the second GMM test H0 : δ = 0N given α = 0N , we dene gt = R1t ⊗ εt. Thedierent moments are given by:

gT (B) =1

T

T∑t=1

R1t ⊗ (R2t −B′R1t).

The GMM estimator of B under the assumption α = 0N is:

B = Y ′R1(R′1R1)

−1.

We now have δ = B1K − 1N .

Then, the GMM test is:

Wa2 = T δ′(

(1′K(V11 + µ1µ′1)−1 ⊗ IN)ST (V11 + µ1µ

′1)−11K ⊗ IN)

)−1δ ∼ χ2

N

where ST is the autocorrelation consistent estimator presented above, but with the newdenitions of gt and the estimator B of B. To improve nite sample properties, we use thefollowing F-test:

Fa2 =

(T −K −N − 1

N

)Wa2

T∼ FN,T−K−N−1.

32

B Dynamic Conditional Correlations (DCC)

To build our DCC model, we employ the methodology described in Engle and Sheppard(2001). The model for N assets is dened as:

rt = µt + at

at = Ht1/2zt

Ht = DtRtDt

where

rt: N x 1 vector of returns at t.

at: N x 1 vector of mean-corrected returns at t, E[at] = 0, Cov[at] = Ht.

µt: N x 1 vector of the conditional expected value of rt.

Ht: N x N matrix of the conditional variance at at t.

Dt: N x N diagonal matrix of conditional standard deviations of at at t.

Rt: N x N correlation matrix of the standardized residuals εt at t (univariate GARCH).

zt: N x 1 vector of i.i.d. normal distribution E[zt] = 0 and E[ztzTt ] = I.

To control for the autocorrelation found in our data, we assume µt to follow an ARMAprocess:

rt = µ+

p∑i=1

φirt−1 +

q∑j=1

θiat−1.

Elements on the diagonal matrix Dt are the standard deviations estimated by dierent uni-variate GARCH models. A GARCH(1,1) will be used.

Dt =

√h1t 0 . . . 0

0√h2t

. . ....

.... . . . . . 0

0 . . . 0√hnt

with

hit = αi0 + αi1a2i,t−1 + βi1hi,t−1.

εt are the standardized residuals so:

εt = Dt−1at ∼ N(0,Rt)

33

Rt =

1 ρ12,t ρ13,t . . . ρ1n,t

ρ12,t 1 ρ23,t. . . ρ2n,t

......

. . . . . . ρn−1,n,tρ1n,t ρ2n,t . . . ρn−1,n,t 1

.As Ht = DtRtDt, the elements are:

[Ht]ij =√hithjtρij.

To ensure that the matrices Ht and Rt are positive denite and that the elements of matrixRt are less than or equal to 1 in absolute value, Rt is decomposed as:

Rt = diag[Qt]−1Qtdiag[Qt]

−1

Qt = (1− δ − ξ)Q+ δεt−1εt−1T + ξQt−1

where

Q =1

T

T∑t=1

εtεtT

and the parameters δ and ξ are scalars. Once again, to ensure that the matrices are positivedenite, it is necessary that:

δ ≥ 0, ξ ≥ 0, δ + ξ < 1.

For the maximum likelihood estimation, we assume the standard errors zt as i.i.d. normaldistribution with E[zt] = 0 and E[ztz

tt ] = I, therefore:

f(zt) =T∏t=1

1

(2π)n/2e−

12zTt zt .

Given that at = Ht1/2zt, we obtain the following likelihood function:

L(θ) =T∏t=1

1

(2π)n/2|Ht|1/2e−

12atTHt

−1at

34

where θ are the parameters to estimate. The log-likelihood function can be written as:

ln(L(θ)) = −1

2

T∑t=1

(nln(2π) + ln(|Ht|) + at

THt−1at

)= −1

2

T∑t=1

(nln(2π) + ln(|DtRtDt|+ at

TDt−1Rt

−1Dt−1at

)− 1

2

T∑t=1

(nln(2π) + 2ln(|Dt|) + ln(|Rt|) + at

TDt−1Rt

−1Dt−1at

).

The rst part is considered as the volatility component where the parameters to be estimatedare those from the univariate models. The second part is the correlation component wherethe parameters used are those obtained from univariate models in addition to δ and ξ. Toease the estimation, Engle (2002) propose a two-step procedure, as follows, 1- use the rstterm to estimate the univariate parameters and then 2- with these univariate parameters,estimate δ and ξ by means of the complete log-likelihood function.

C Implementation First-Order Stochastic Dominance

To ease computations, Scaillet and Topaloglou (2010) propose the following steps:

1.√T/T can be left out as T is xed.

2. We can determine a number T of values, R = r1, r2, . . . , rT, holding the optimal valueof z, where R is the benchmark portfolio τ ′Yt vector of returns.

3. We solve several problems P(r), r ∈ R and take the value that maximizes J1(z, τ, F )−J1(z, λ, F ). By setting z=r, the optimal values of Lt are therefore known.

4. We dene Mt = minYt,i, i=1, . . . , n, which is the minimum of vector Yt at t.

The P(r) problem to be solved for the set of r values is given by:

minλ

T∑t=1

Wt

s.t. λ′Yt ≥ r − (r −Mt)Wt, ∀t ∈ T∑λ = 1

Wt ∈ 0, 1, ∀t ∈ T.

35

D Implementation Second-Order Stochastic Dominance

We outline its complete formulation below:

maxz,λ

S2 =√T

1

T

T∑t=1

(Lt −Wt)

s.t. M(Ft − 1) ≤ z − τ ′Yt ≤MFt, ∀t ∈ T−M(1− Ft) ≤ Lt − (z − τ ′Yt) ≤M(1− Ft), ∀t ∈ T−MFt ≤ Lt ≤MFt, ∀t ∈ TWt ≥ z − λ′Yt, ∀t ∈ T∑

λ = 1

Wt ≥ 0, Ft ∈ 0, 1, ∀t ∈ T

with M being a large constant. 1T

∑Tt=1 Lt and

1T

∑Tt=1Wt represent J2(z, τ, F ) and J2(z, λ, F )

respectively. Ft is a binary variable equal to 1 when z ≥ τ ′Yt and 0 otherwise. Following thesame steps as 3.4.2, the problem can be transformed into the following linear programmingP(r) problem:

minλ

T∑t=1

Wt

s.t. Wt ≥ r − λ′Yt, ∀t ∈ T∑λ = 1

Wt ≥ 0, ∀t ∈ T.

The procedure can now be readily solved in less than a minute.

36

Table 1Indexes

This table describes the indexes used in the study. Panel A shows those for Cat Bonds, Panel B for asset classes and Panel C for factors

Panel A: Cat Bonds

Name DescriptionSwiss Re Cat Bond Index The index tracks the total returns of Cat Bonds denominated in USDSwiss Re Global Cat Bond Index The index tracks the overall performance of Cat Bonds denominated

in USD, EUR and JPYSwiss Re BB Cat Bond Index The index tracks total returns of USD denominated Cat Bonds rated

BBSwiss Re Global Unhedged Cat Bond Index The index tracks the overall performance of Cat Bonds denominated

in USD, EUR and JPY. It captures the total impact of currency riskSwiss Re US Wind Cat Bond Index The index tracks total returns of Cat Bonds associated with US winds

Panel B: Asset Classes

Class NameUS Equities US - Datastream MarketGlobal Equities ex-US MSCI World ex-USAEmerging Markets Equities MSCI Emerging MarketsReal-Estate US - Datastream REITsUS Treasury Bonds Bloomberg Barclays US TreasuryUS Corporate Bonds Bloomberg Barclays US Corporate BondsUS High-Yield Bonds Bloomberg Barclays US High-Yield BondsCommodities S&P Goldman Sachs Commodity

Panel C: Factors

Factor Long position Short positionValue MSCI US Value MSCI US GrowthSize MSCI US Small Cap MSCI US Large CapMomentum MSCI US Momentum MSCI USVolatility MSCI US Minimum Volatility MSCI USMortgage Barclays US MBS Barclays US Treasury IntermediateDefault Bloomberg Barclays US Corporate Bonds Bloomberg Barclays US Treasury BondsTerm Bloomberg Barclays US Treasury 10-20yrs Bloomberg Barclays US Short TreasuryHigh-Yield Bloomberg Barclays US High-Yield Bonds Bloomberg Barclays US Corporate BondsCommodity Curve S&P GSCI Dynamic Roll S&P Goldman Sachs CommodityEquity Market MSCI US 1 Month T-Bill Rate

37

Table 2Descriptive statistics

The table presents the descriptive statistics of weekly returns Rt = ln(St+1

St

)of the various indexes for the full sample (January 4, 2002 - June 30, 2017). The rst column is the weekly average

of the returns µ = 1T

∑Tt=1Rt. The second column is the standard deviation σ =

√1T

∑Tt=1[Rt − µ]2. The third column is the Sharpe ratio SR =

µ−rfσ

where rf is the average risk-free rate

(US 1 month T-Bill). The fourth column the skewness Sk = 1T

∑Tt=1

([Rt−µ]3σ3

). The fth the kurtosis Ku = 1

T

∑Tt=1

([Rt−µ]4σ4

). The sixth column is the p-value of the Jarque-Bera test

where the null hypothesis is the normality of returns JB = T6

(Sk2 +

[Ku−3]2

4

)∼ χ2(2). The table presents next, the date of the minimum return (Dt Min), the minimum (Min), the date

of the maximum (Dt Max), the maximum (Max), the number of negative returns (<0%), the number of returns under -5% (<-5%) in percent, the number of returns under -10% (<-10%) inpercent and the maximum drawdown (MDD) (dened as the decrease between a high and a low recorded over a specic period expressed as a percentage) respectively.

µ σ SR Sk Ku JB -pvalue

Dt Min Min Dt Max Max <0% <-5% <-10% MDD

SRCAT 0.15% 0.36% 0.349 -1.4 39.31 0.00% 2004/09/10 -3.24% 2004/09/17 3.94 % 11% 0% 0% 6.69%SRGLCAT 0.14% 0.33% 0.368 -1.59 40.2 0.00% 2004/09/10 -3.01% 2004/09/17 3.66 % 11% 0% 0% 6.16%SRBBCAT 0.12% 0.39% 0.254 -2.65 50.31 0.00% 2004/09/10 -4.09% 2004/09/17 4.26 % 12% 0% 0% 7.51%SRGLUCAT 0.14% 0.36% 0.335 -1.21 28.94 0.00% 2004/09/10 -2.89% 2004/09/17 3.62 % 19% 0% 0% 6.38%SRUSWIND 0.16% 0.72% 0.192 -2.57 159.41 0.00% 2004/09/10 -11.89% 2004/09/17 10.58 % 16% 0% 0% 16.77%USEQ 0.13% 2.37% 0.047 -0.88 8.61 0.00% 2008/10/10 -20.01% 2008/11/28 11.81 % 43% 2% 0% 27.13%XUSEQ 0.13% 2.59% 0.042 -1.5 11.67 0.00% 2008/10/10 -24.38% 2008/11/28 11.34 % 44% 3% 1% 30.14%EMEQ 0.19% 3.08% 0.054 -0.81 7.61 0.00% 2008/10/10 -22.51% 2008/10/31 18.67 % 43% 4% 1% 31.22%RE 0.19% 3.40% 0.05 -0.44 8.7 0.00% 2008/11/21 -19.82% 2008/11/28 20.53 % 43% 5% 1% 36.99%USTB 0.08% 0.60% 0.095 -0.28 0.72 0.00% 2009/06/05 -2.05% 2008/11/21 2.37 % 43% 0% 0% 4.41%USCB 0.11% 0.72% 0.116 -0.8 4.85 0.00% 2008/09/19 -4.89% 2008/12/19 3.41 % 41% 0% 0% 6.61%USHY 0.15% 1.06% 0.124 -2.39 25.95 0.00% 2008/10/10 -11.30% 2009/01/09 5.76 % 33% 0% 0% 13.41%COMM -0.04% 3.29% -0.018 -0.81 3.28 0.00% 2008/12/05 -21.13% 2009/01/02 12.13 % 47% 6% 1% 30.09%VALUE -0.01% 0.98% -0.010 0.10 4.55 0.00% 2002/07/26 -5.85% 2009/05/08 5.79 % 53% 0% 0% 10.04%SIZE 0.04% 1.25% 0.035 -0.23 0.91 0.00% 2008/10/24 -4.53% 2008/11/28 4.58 % 46% 0% 0% 8.83%MOM 0.05% 1.01% 0.049 -0.61 3.25 0.00% 2008/07/18 -5.20% 2002/07/26 4.50 % 45% 0% 0% 9.71%VOL 0.02% 0.81% 0.023 0.19 1.36 0.00% 2002/05/17 -3.08% 2008/10/03 3.67 % 50% 0% 0% 6.09%MORTG 0.02% 0.26% 0.062 -0.4 4.61 0.00% 2008/03/07 -1.67% 2008/09/12 1.19 % 47% 0% 0% 2.47%DEFAULT 0.03% 0.46% 0.059 -2.19 23.14 0.00% 2008/09/19 -4.80% 2009/05/08 1.97 % 41% 0% 0% 5.79%TERM 0.09% 1.18% 0.077 -0.2 0.93 0.00% 2009//01/23 -4.10% 2008/11/21 5.32 % 44% 0% 0% 9.42%HY 0.05% 1.04% 0.046 -1.16 10.04 0.00% 2008/11/21 -8.32% 2009/01/02 5.49 % 46% 0% 0% 10.93%C.CURVE 0.16% 1.12% 0.146 0.2 8.37 0.00% 2008/12/12 -6.59% 2008/12/05 8.79 % 43% 0% 0% 15.38%EQ 0.11% 2.37% 0.045 -0.86 8.55 0.00% 2008/10/10 -20.05% 2008/11/28 11.58 % 44% 2% 0% 27.29%

Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: Swiss Re Global Cat Bond Index, SRBBCAT: Swiss Re BB Cat Bond Index, SRGLUCAT: Swiss Re Global Unhedged Cat BondIndex, SRUSWIND: Swiss Re US Wind Cat Bond Index. Asset Classes - USEQ: US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCI World ex-USA), EMEQ:Emerging Markets Equities (MSCI Emerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US Treasury Bonds (Bloomberg Barclays US Treasury), USCB: US Corporate Bonds(Bloomberg Barclays US Corporate Bonds), USHYB: US High-Yield Bonds (Bloomberg Barclays US High-Yield Bonds), COMM: Commodities (S & P Goldman Sachs Commodity). Factors- VALUE: Value Factor (Long: MSCI US Value - Short: MSCI US Growth), SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap), MOM: Momentum Factor (Long:MSCI US Momentum - Shortt: MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility - Short: MSCI US), MORTG: Mortgage Factor (Long: Barclays US MBS - Short:MSCI US), DEFAULT: Default Factor (Long: Bloomberg Barclays US Corporate Bonds - Short: Bloomberg Barclays US Treasury Bonds), TERM: Term Factor (Long: Bloomberg BarclaysUS Treasury 10-20y - Short: Bloomberg Barclays US Short Treasury), HY: High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short: Bloomberg Barclays US CorporateBonds), C.Curve: Commodity Curve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P Goldman Sachs Commodity), EQ: Equity Market Factor (Long: MSCI US - Short: 1 MonthT-Bill).

38

Table 3Descriptive statistics - Financial crisis (12/2007 - 05/2009)

The table presents the descriptive statistics of weekly returns Rt = ln(St+1

St

)of the various indexes during the 2008 nancial crisis (December 2007 - May 2009). The rst column is the

weekly average of the returns µ = 1T

∑Tt=1Rt. The second column is the standard deviation σ =

√1T

∑Tt=1[Rt − µ]2. The third column is the Sharpe ratio SR =

µ−rfσ

where rf is the

average risk-free rate (US 1 month T-Bill). The fourth column the skewness Sk = 1T

∑Tt=1

([Rt−µ]3σ3

). The fth the kurtosis Ku = 1

T

∑Tt=1

([Rt−µ]4σ4

). The sixth column is the p-value of the

Jarque-Bera test where the null hypothesis is the normality of returns JB = T6

(Sk2 +

[Ku−3]2

4

)∼ χ2(2). The table presents next, the date of the minimum return (Dt Min), the minimum

(Min), the date of the maximum (Dt Max), the maximum (Max), the number of negative returns (<0%), the number of returns under -5% (<-5%) in percent, the number of returns under-10% (<-10%) in percent and the maximum drawdown (MDD) (dened as the decrease between a high and a low recorded over a specic period expressed as a percentage) respectively.

µ σ SR Sk Ku <0% <-5% <-10%

SRCAT 0.06% 0.41% 0.113 -3.69 18.14 18% 0% 0%SRGLCAT 0.07% 0.41% 0.123 -4.21 22.44 15% 0% 0%SRBBCAT 0.05% 0.50% 0.057 -4.18 21.98 18% 0% 0%SRGLUCAT 0.07% 0.48% 0.099 -2.98 14.81 32% 0% 0%SRUSWIND 0.03% 0.42% 0.035 -3.28 15.33 22% 0% 0%USEQ -0.55% 4.65% -0.123 -0.58 3.23 54% 12% 3%XUSEQ -0.62% 5.14% -0.124 -1.25 4.53 54% 17% 4%EMEQ -0.55% 5.97% -0.096 -0.45 2.86 54% 18% 5%RE -0.77% 7.69% -0.102 0 0.95 54% 21% 12%USTB 0.11% 0.82% 0.115 0.05 0 46% 0% 0%USCB 0.00% 1.25% -0.011 -1.04 3.29 49% 0% 0%USHY -0.08% 2.46% -0.04 -1.44 5.8 47% 4% 1%COMM -0.66% 5.60% -0.12 -0.82 1.5 47% 21% 6%VALUE -0.09% 1.84% -0.049 0.5 0.56 59% 0% 0%SIZE 0.07% 1.81% 0.038 -0.47 -0.09 41% 0% 0%MOM -0.13% 1.82% -0.069 -0.45 0.25 47% 1% 0%VOL 0.11% 0.96% 0.115 0.46 1.21 42% 0% 0%MORTG 0.03% 0.55% 0.051 -0.47 0.28 46% 0% 0%DEFAULT -0.11% 1.09% -0.098 -1.29 4.1 54% 0% 0%TERM 0.07% 1.68% 0.039 0.29 0.85 46% 0% 0%HY -0.09% 2.14% -0.04 -0.96 2.99 47% 4% 0%C.CURVE 0.43% 2.14% 0.199 0.1 3.93 31% 3% 0%EQ -0.58% 4.64% -0.124 -0.57 3.23 54% 12% 3%

Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: Swiss Re Global Cat Bond Index, SRBBCAT: Swiss Re BB Cat Bond Index, SRGLUCAT: Swiss Re Global Unhedged Cat BondIndex, SRUSWIND: Swiss Re US Wind Cat Bond Index. Asset Classes - USEQ: US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCI World ex-USA), EMEQ:Emerging Markets Equities (MSCI Emerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US Treasury Bonds (Bloomberg Barclays US Treasury), USCB: US Corporate Bonds(Bloomberg Barclays US Corporate Bonds), USHYB: US High-Yield Bonds (Bloomberg Barclays US High-Yield Bonds), COMM: Commodities (S & P Goldman Sachs Commodity). Factors- VALUE: Value Factor (Long: MSCI US Value - Short: MSCI US Growth), SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap), MOM: Momentum Factor (Long:MSCI US Momentum - Shortt: MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility - Short: MSCI US), MORTG: Mortgage Factor (Long: Barclays US MBS - Short:MSCI US), DEFAULT: Default Factor (Long: Bloomberg Barclays US Corporate Bonds - Short: Bloomberg Barclays US Treasury Bonds), TERM: Term Factor (Long: Bloomberg BarclaysUS Treasury 10-20y - Short: Bloomberg Barclays US Short Treasury), HY: High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short: Bloomberg Barclays US CorporateBonds), C.Curve: Commodity Curve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P Goldman Sachs Commodity), EQ: Equity Market Factor (Long: MSCI US - Short: 1 MonthT-Bill).

39

Table 4Correlation matrix - Cat Bonds and Asset Classes

Panel A presents the correlations between weekly Cat Bonds returns and asset class returns for the full sample (January 4, 2002 - June 30, 2017). Panel B presents the correlations betweenCat Bonds and factors for the same period. Note that because of the strong correlation between the various Cat Bonds indexes, only the correlations of factors with the SRCAT index arepresented.

Panel A: Cat Bonds and Asset Classes

SRCAT SRGLCAT SRBBCAT SRGLUCATSRUSWND USEQ XUSEQ EMEQ RE USTB USCB USHYB COMM

SRCAT 1 0.997 0.942 0.931 0.800 0.075 0.115 0.103 0.057 0.032 0.154 0.151 0.097SRGLCAT 0.997 1 0.941 0.936 0.792 0.093 0.134 0.117 0.061 0.030 0.160 0.172 0.106SRBBCAT 0.942 0.941 1 0.874 0.779 0.079 0.108 0.105 0.053 0.045 0.174 0.157 0.105SRGLUCAT 0.931 0.936 0.874 1 0.731 0.176 0.286 0.257 0.141 0.032 0.180 0.230 0.213SRUSWND 0.800 0.792 0.779 0.731 1 0.027 0.053 0.069 0.036 0.021 0.091 0.083 0.051USEQ 0.075 0.093 0.079 0.176 0.027 1 0.83 0.72 0.70 -0.41 -0.10 0.55 0.32XUSEQ 0.115 0.134 0.108 0.286 0.053 0.826 1 0.861 0.571 -0.290 0.050 0.595 0.452EMEQ 0.103 0.117 0.105 0.257 0.069 0.724 0.861 1 0.547 -0.258 0.078 0.576 0.451RE 0.057 0.061 0.053 0.141 0.036 0.705 0.571 0.547 1 -0.182 0.038 0.444 0.195USTB 0.032 0.030 0.045 0.032 0.021 -0.410 -0.290 -0.258 -0.182 1 0.776 -0.130 -0.176USCB 0.154 0.160 0.174 0.180 0.091 -0.102 0.050 0.078 0.038 0.776 1 0.364 -0.024USHYB 0.151 0.172 0.157 0.230 0.083 0.549 0.595 0.576 0.444 -0.130 0.364 1 0.304COMM 0.097 0.106 0.105 0.213 0.051 0.321 0.452 0.451 0.195 -0.176 -0.024 0.304 1

Panel B: Cat Bonds and Factors

SRCAT SRUSWIND VALUE SIZE MOM VOL MORTG DEFAULT TERM HY C.CURVE EQ

SRCAT 1 0.800 0.009 -0.024 -0.010 0.003 0.018 0.200 0.037 0.048 -0.065 0.073SRUSWIND 0.800 1 0.036 -0.035 0.009 0.021 0.009 0.115 0.035 0.022 -0.034 0.025VALUE 0.009 0.036 1 -0.066 -0.374 0.091 0.120 0.083 -0.030 0.118 -0.113 0.133SIZE -0.024 -0.035 -0.066 1 0.115 -0.378 0.129 0.197 -0.209 0.362 -0.102 0.379MOM -0.010 0.009 -0.374 0.115 1 0.258 -0.101 -0.115 0.180 -0.164 0.066 -0.190VOL 0.003 0.021 0.091 -0.378 0.258 1 -0.297 -0.269 0.488 -0.486 0.192 -0.627MORTG 0.018 0.009 0.120 0.129 -0.101 -0.297 1 0.351 -0.275 0.387 -0.138 0.396DEFAULT 0.200 0.115 0.083 0.197 -0.115 -0.269 0.351 1 -0.041 0.373 -0.111 0.376TERM 0.037 0.035 -0.030 -0.209 0.180 0.488 -0.275 -0.041 1 -0.659 0.157 -0.396HY 0.048 0.022 0.118 0.362 -0.164 -0.486 0.387 0.373 -0.659 1 -0.220 0.622C.CURVE -0.065 -0.034 -0.113 -0.102 0.066 0.192 -0.138 -0.111 0.157 -0.220 1 -0.206EQ 0.073 0.025 0.133 0.379 -0.190 -0.627 0.396 0.376 -0.396 0.622 -0.206 1

Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: Swiss Re Global Cat Bond Index, SRBBCAT: Swiss Re BB Cat Bond Index, SRGLUCAT: Swiss Re Global Unhedged Cat BondIndex, SRUSWIND: Swiss Re US Wind Cat Bond Index. Asset Classes - USEQ: US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCI World ex-USA), EMEQ:Emerging Markets Equities (MSCI Emerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US Treasury Bonds (Bloomberg Barclays US Treasury), USCB: US Corporate Bonds(Bloomberg Barclays US Corporate Bonds), USHYB: US High-Yield Bonds (Bloomberg Barclays US High-Yield Bonds), COMM: Commodities (S & P Goldman Sachs Commodity). Factors- VALUE: Value Factor (Long: MSCI US Value - Short: MSCI US Growth), SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap), MOM: Momentum Factor (Long:MSCI US Momentum - Short: MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility - Short: MSCI US), MORTG: Mortgage Factor (Long: Barclays US MBS - Short:MSCI US), DEFAULT: Default Factor (Long: Bloomberg Barclays US Corporate Bonds - Short: Bloomberg Barclays US Treasury Bonds), TERM: Term Factor (Long: Bloomberg BarclaysUS Treasury 10-20y - Short: Bloomberg Barclays US Short Treasury), HY: High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short: Bloomberg Barclays US CorporateBonds), C.Curve: Commodity Curve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P Goldman Sachs Commodity), EQ: Equity Market Factor (Long: MSCI US - Short: 1 MonthT-Bill).

40

Table 5Correlation matrix (Financial crisis: 12/2007 - 05/2009)

Panel A presents the correlations between weekly Cat Bonds returns and asset class returns during the last nancial crisis (December 2007 - Mai 2009). Panel B presents the correlationsbetween Cat Bonds and factors for the same period. Note that because of the strong correlation between the various Cat Bonds indexes only the correlations of factors with the SRCAT indexare presented.

Panel A: Cat Bonds and Asset Classes

SRCAT SRGLCAT SRBBCAT SRGLUCATSRUSWND USEQ XUSEQ EMEQ RE USTB USCB USHYB COMM

SRCAT 1 0.992 0.979 0.856 0.913 0.284 0.370 0.352 0.073 0.041 0.508 0.430 0.323SRGLCAT 0.992 1 0.971 0.869 0.917 0.338 0.423 0.388 0.083 0.039 0.513 0.472 0.339SRBBCAT 0.979 0.971 1 0.837 0.897 0.279 0.348 0.332 0.077 0.069 0.537 0.400 0.287SRGLUCAT 0.856 0.869 0.837 1 0.801 0.447 0.630 0.603 0.222 0.078 0.491 0.503 0.533SRUSWND 0.913 0.917 0.897 0.801 1 0.322 0.405 0.376 0.111 0.083 0.488 0.427 0.326USEQ 0.284 0.338 0.279 0.447 0.322 1 0.857 0.815 0.775 -0.347 0.124 0.693 0.457XUSEQ 0.370 0.423 0.348 0.630 0.405 0.857 1 0.914 0.585 -0.230 0.267 0.721 0.609EMEQ 0.352 0.388 0.332 0.603 0.376 0.815 0.914 1 0.646 -0.273 0.239 0.647 0.611RE 0.073 0.083 0.077 0.222 0.111 0.775 0.585 0.646 1 -0.314 0.036 0.500 0.257USTB 0.041 0.039 0.069 0.078 0.083 -0.347 -0.230 -0.273 -0.314 1 0.513 -0.277 -0.292USCB 0.508 0.513 0.537 0.491 0.488 0.124 0.267 0.239 0.036 0.513 1 0.493 0.020USHYB 0.430 0.472 0.400 0.503 0.427 0.693 0.721 0.647 0.500 -0.277 0.493 1 0.413COMM 0.323 0.339 0.287 0.533 0.326 0.457 0.609 0.611 0.257 -0.292 0.020 0.413 1

Panel B: Cat Bonds and Factors

SRCAT SRUSWIND VALUE SIZE MOM VOL MORTG DEFAULT TERM HY C.CURVE EQ

SRCAT 1 0.913 -0.034 -0.110 -0.169 -0.095 0.031 0.554 0.065 0.196 -0.143 0.288SRUSWIND 0.913 1 0.098 -0.050 -0.197 -0.123 0.035 0.499 0.096 0.205 -0.160 0.324VALUE -0.034 0.098 1 -0.060 -0.397 0.099 0.177 0.006 0.004 0.175 -0.171 0.382SIZE -0.110 -0.050 -0.060 1 -0.047 -0.476 0.158 0.217 -0.212 0.427 -0.144 0.459MOM -0.169 -0.197 -0.397 -0.047 1 0.144 -0.101 -0.103 0.058 -0.216 0.231 -0.380VOL -0.095 -0.123 0.099 -0.476 0.144 1 -0.269 -0.327 0.306 -0.463 0.314 -0.588MORTG 0.031 0.035 0.177 0.158 -0.101 -0.269 1 0.389 -0.201 0.388 -0.305 0.357DEFAULT 0.554 0.499 0.006 0.217 -0.103 -0.327 0.389 1 -0.082 0.441 -0.122 0.402TERM 0.065 0.096 0.004 -0.212 0.058 0.306 -0.201 -0.082 1 -0.580 0.400 -0.290HY 0.196 0.205 0.175 0.427 -0.216 -0.463 0.388 0.441 -0.580 1 -0.351 0.719C.CURVE -0.143 -0.160 -0.171 -0.144 0.231 0.314 -0.305 -0.122 0.400 -0.351 1 -0.332EQ 0.288 0.324 0.382 0.459 -0.380 -0.588 0.357 0.402 -0.290 0.719 -0.332 1

Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: Swiss Re Global Cat Bond Index, SRBBCAT: Swiss Re BB Cat Bond Index, SRGLUCAT: Swiss Re Global Unhedged Cat BondIndex, SRUSWIND: Swiss Re US Wind Cat Bond Index. Asset Classes - USEQ: US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCI World ex-USA), EMEQ:Emerging Markets Equities (MSCI Emerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US Treasury Bonds (Bloomberg Barclays US Treasury), USCB: US Corporate Bonds(Bloomberg Barclays US Corporate Bonds), USHYB: US High-Yield Bonds (Bloomberg Barclays US High-Yield Bonds), COMM: Commodities (S & P Goldman Sachs Commodity). Factors- VALUE: Value Factor (Long: MSCI US Value - Short: MSCI US Growth), SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap), MOM: Momentum Factor (Long:MSCI US Momentum - Short: MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility - Short: MSCI US), MORTG: Mortgage Factor (Long: Barclays US MBS - Short:MSCI US), DEFAULT: Default Factor (Long: Bloomberg Barclays US Corporate Bonds - Short: Bloomberg Barclays US Treasury Bonds), TERM: Term Factor (Long: Bloomberg BarclaysUS Treasury 10-20y - Short: Bloomberg Barclays US Short Treasury), HY: High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short: Bloomberg Barclays US CorporateBonds), C.Curve: Commodity Curve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P Goldman Sachs Commodity), EQ: Equity Market Factor (Long: MSCI US - Short: 1 MonthT-Bill).

41

Table 6Mean-variance spanning tests (Full sample 2002-2017)

This table reports the results of the various mean-variance spanning tests from the addition of Cat Bonds to a universe of asset classes (PanelA) or factors (Panel B) for the full sample (January 4, 2002 - June 30, 2017). The rst column (GMM) reports the F statistics as well asthe corresponding p-values. The null hypothesis is that the mean-variance frontier constructed from the universe of asset classes (Panel A) orfactors (Panel B) is not signicantly dierent from the one constructed with the augmented universe. The second column (SD1) presents the Fstatistics and p-values of the rst step-down test of Kan and Zhou (2012). The null hypothesis is that the tangent portfolio constructed fromthe base universe is not signicantly dierent from the one with Cat Bonds included. The third column (SD2) presents the F statistics andp-values of the second step-down test of Kan and Zhou (2012). The null hypothesis is that the minimum-variance portfolio constructed fromthe base universe is not signicantly dierent from the one with Cat Bonds included.

Panel A: Asset Classes

GMM SD1 SD2SRCAT 495.49*** 77.39*** 602.08***

(0.00%) (0.00%) (0.00%)SRGLCAT 384.16*** 82.38*** 484.45***

(0.00%) (0.00%) (0.00%)SRBBCAT 349.79*** 41.44*** 343.28***

(0.00%) (0.00%) (0.00%)SRGLUCAT 217.38*** 80.51*** 552.99***

(0.00%) (0.00%) (0.00%)SRUSWIND 155.14*** 101.51*** 319.81***

(0.00%) (0.00%) (0.00%)

Panel B: Factors

GMM SD1 SD2SRCAT 173.11*** 83.14*** 162.97***

(0.00%) (0.00%) (0.00%)SRGLCAT 187.35*** 100.43*** 202.98***

(0.00%) (0.00%) (0.00%)SRBBCAT 128.52*** 45.80*** 137.76***

(0.00%) (0.00%) (0.00%)SRGLUCAT 130.88*** 81.99*** 146.54***

(0.00%) (0.00%) (0.00%)SRUSWIND 50.23*** 75.19*** 24.10***

(0.00%) (0.00%) (0.00%)

*Null hypothesis rejected at a level >90%, **Null hypothesis rejected a level >95%, ***Null hypothesis rejected at a level >99%.

Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: Swiss Re Global Cat Bond Index, SRBBCAT: Swiss Re BB Cat Bond Index,SRGLUCAT: Swiss Re Global Unhedged Cat Bond Index, SRUSWIND: Swiss Re US Wind Cat Bond Index. Asset Classes - USEQ:US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCI World ex-USA), EMEQ: Emerging Markets Equities (MSCIEmerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US Treasury Bonds (Bloomberg Barclays US Treasury), USCB:US Corporate Bonds (Bloomberg Barclays US Corporate Bonds), USHYB: US High-Yield Bonds (Bloomberg Barclays US High-Yield Bonds),COMM: Commodities (S & P Goldman Sachs Commodity). Factors - VALUE: Value Factor (Long: MSCI US Value - Short: MSCI US Growth),SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap), MOM: Momentum Factor (Long: MSCI US Momentum - Short:MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility - Short: MSCI US), MORTG: Mortgage Factor (Long: BarclaysUS MBS - Short: MSCI US), DEFAULT: Default Factor (Long: Bloomberg Barclays US Corporate Bonds - Short: Bloomberg Barclays USTreasury Bonds), TERM: Term Factor (Long: Bloomberg Barclays US Treasury 10-20y - Short: Bloomberg Barclays US Short Treasury), HY:High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short: Bloomberg Barclays US Corporate Bonds), C.Curve: CommodityCurve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P Goldman Sachs Commodity), EQ: Equity Market Factor (Long: MSCI US -Short: 1 Month T-Bill).

42

Table 7Mean-variance spanning tests (Regimes from a Markov-Switching model)

This table reports the results of the various mean-variance tests from the addition of Cat Bonds to a universe of asset classes (Panel A) orfactors (Panel B) for regimes built from a Markov-Switching Hardy (2001) on the US bond index (Bloomberg Barclays US Aggregate Bond)and for those built from a similar on US stock index (S&P500). For each regime, there are three columns. The rst column (GMM) reports theF statistics as well as the corresponding p-values. The null hypothesis is that the mean-variance frontier constructed from the universe of assetclasses (Panel A) or factors (Panel B) is not signicantly dierent from the one constructed with the augmented universe. The second column(SD1) presents the F statistics and p-values of the rst step-down test of Kan and Zhou (2012). The null hypothesis is that the tangent portfolioconstructed from the base universe is not signicantly dierent from the one with Cat Bonds included. The third column (SD2) presents theF statistics and p-values of the second step-down test of Kan and Zhou (2012). The null hypothesis is that the minimum-variance portfolioconstructed from the base universe is not signicantly dierent from the one with Cat Bonds included.

Panel A: Asset Classes

Markov Switching (US aggregate bond) Markov Switching (S&P 500)Low volatility High volatility Low volatility High volatility

GMM SD1 SD2 GMM SD1 SD2 GMM SD1 SD2 GMM SD1 SD2SRCAT 364.67*** 102.96*** 379.94*** 292.06*** 18.03*** 581.80*** 499.17*** 54.82*** 483.37*** 174.01*** 25.25*** 301.15***

(0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%)SRGLCAT 426.41*** 106.94*** 442.64*** 257.15*** 19.60*** 512.75*** 567.93*** 59.27*** 535.56*** 132.78*** 29.68*** 245.51***

(0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%)SRBBCAT 222.05*** 34.01*** 340.10*** 220.64*** 11.87*** 467.49*** 302.64*** 21.55*** 444.53*** 120.14*** 22.90*** 226.35***

(0.00%) (0.00%) (0.00%) (0.00%) (0.07%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%)SRGLU 299.06*** 77.99*** 310.47*** 118.97*** 15.94*** 288.92*** 358.85*** 59.65*** 440.78*** 78.66*** 34.12*** 210.80***

(0.00%) (0.00%) (0.00%) (0.00%) (0.01%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%)SRUSWD 103.54*** 81.07*** 79.75*** 164.82*** 8.44*** 347.31*** 129.60*** 83.07*** 139.54*** 68.82*** 11.43*** 109.87***

(0.00%) (0.00%) (0.00%) (0.00%) (0.42%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.09%) (0.00%)

Panel B: Factors

Markov Switching (US aggregate bond) Markov Switching (S&P 500)Low volatility High volatility Low volatility High volatility

GMM SD1 SD2 GMM SD1 SD2 GMM SD1 SD2 GMM SD1 SD2SRCAT 54.72*** 66.63*** 29.71*** 80.31*** 26.62*** 152.05*** 108.40*** 53.00*** 82.91*** 60.06*** 32.23*** 58.61***

(0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%)SRGLCAT 65.47*** 75.01*** 39.23*** 108.40*** 28.35*** 187.23*** 121.28*** 57.79*** 89.80*** 73.73*** 36.92*** 74.17***

(0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%)SRBBCAT 31.78*** 33.91*** 23.96*** 49.87*** 17.57*** 117.40*** 104.89*** 23.35*** 115.11*** 44.83*** 27.68*** 36.57***

(0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%)SRGLU 65.28*** 109.53*** 29.22*** 40.35*** 31.26*** 72.11*** 81.43*** 58.39*** 67.43*** 38.25*** 39.16*** 57.84***

(0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%)SRUSWD 44.76*** 73.58*** 2.20 44.70*** 11.50*** 88.93*** 60.49*** 59.59*** 36.63*** 8.80*** 13.75*** 6.46**

(0.00%) (0.00%) (13.89%) (0.00%) (0.09%) (0.00%) (0.00%) (0.00%) (0.00%) (0.02%) (0.03%) (1.18%)

*Null hypothesis rejected at a level >90%, **Null hypothesis rejected a level >95%, ***Null hypothesis rejected at a level >99%.

Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: Swiss Re Global Cat Bond Index, SRBBCAT: Swiss Re BB Cat Bond Index,SRGLUCAT: Swiss Re Global Unhedged Cat Bond Index, SRUSWIND: Swiss Re US Wind Cat Bond Index. Asset Classes - USEQ:US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCI World ex-USA), EMEQ: Emerging Markets Equities (MSCIEmerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US Treasury Bonds (Bloomberg Barclays US Treasury), USCB:US Corporate Bonds (Bloomberg Barclays US Corporate Bonds), USHYB: US High-Yield Bonds (Bloomberg Barclays US High-Yield Bonds),COMM: Commodities (S & P Goldman Sachs Commodity). Factors - VALUE: Value Factor (Long: MSCI US Value - Short: MSCI US Growth),SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap), MOM: Momentum Factor (Long: MSCI US Momentum - Short:MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility - Short: MSCI US), MORTG: Mortgage Factor (Long: BarclaysUS MBS - Short: MSCI US), DEFAULT: Default Factor (Long: Bloomberg Barclays US Corporate Bonds - Short: Bloomberg Barclays USTreasury Bonds), TERM: Term Factor (Long: Bloomberg Barclays US Treasury 10-20y - Short: Bloomberg Barclays US Short Treasury), HY:High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short: Bloomberg Barclays US Corporate Bonds), C.Curve: CommodityCurve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P Goldman Sachs Commodity), EQ: Equity Market Factor (Long: MSCI US -Short: 1 Month T-Bill).

43

Table 8Mean-variance spanning tests NBER business cycles and Turbulence Index

This table reports the results of the various mean-variance spanning tests from the addition of Cat Bonds to a universe of asset classes (PanelA) or factors (Panel B) for sub-samples built from the NBER business cycles and from the turbulence index Kritzman and Li (2010). For eachsub-sample , there are three dierent columns. The rst column (GMM) reports the F statistics as well as the corresponding p-values. The nullhypothesis is that the mean-variance frontier constructed from the universe of asset classes (Panel A) or factors (Panel B) is not signicantlydierent from the one constructed with the augmented universe. The second column (SD1) presents the F statistics and p-values of the rststep-down test of Kan and Zhou (2012). The null hypothesis is that the tangent portfolio constructed from the base universe is not signicantlydierent from the one with Cat Bonds included. The third column (SD2) presents the F statistics and p-values of the second step-down testof Kan and Zhou (2012). The null hypothesis is that the minimum-variance portfolio constructed from the base universe is not signicantlydierent from the one with Cat Bonds included.

Panel A: Asset Classes

NBER Turbulence IndexNormal Recession Calm Turbulent

GMM SD1 SD2 GMM SD1 SD2 GMM SD1 SD2 GMM SD1 SD2SRCAT 693.68*** 106.43*** 605.99*** 104.21*** 5.48** 217.03*** 339.96*** 78.65*** 331.97*** 180.56*** 19.30*** 335.36***

(0.00%) (0.00%) (0.00%) (0.00%) (2.22%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%)SRGLCAT 834.02*** 131.93*** 712.66*** 72.59*** 6.70** 192.99*** 397.63*** 86.91*** 385.68*** 151.84*** 22.86*** 263.82***

(0.00%) (0.00%) (0.00%) (0.00%) (1.17%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%)SRBBCAT 452.60*** 47.45*** 619.72*** 121.92*** 2.84* 224.45*** 213.59*** 32.02*** 293.54*** 138.18*** 15.98*** 240.76***

(0.00%) (0.00%) (0.00%) (0.00%) (9.67%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.01%) (0.00%)SRGLU 562.86*** 90.60*** 518.23*** 35.49*** 10.17*** 88.33*** 280.10*** 76.03*** 286.59*** 78.24*** 30.08*** 170.67***

(0.00%) (0.00%) (0.00%) (0.00%) (0.21%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%)SRUSWD 198.98*** 89.07*** 178.27*** 58.25*** 2.18 135.85*** 111.03*** 113.88*** 72.21*** 98.22*** 2.67 179.96***

(0.00%) (0.00%) (0.00%) (0.00%) (14.45%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (10.48%) (0.00%)

Panel B: Factors

NBER Turbulence IndexNormal Recession Calm Turbulent

GMM SD1 SD2 GMM SD1 SD2 GMM SD1 SD2 GMM SD1 SD2SRCAT 86.34*** 80.55*** 57.38*** 41.27*** 8.69*** 104.63*** 113.93*** 75.27*** 81.19*** 38.16*** 12.18*** 72.65***

(0.00%) (0.00%) (0.00%) (0.00%) (0.44%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.07%) (0.00%)SRGLCAT 103.65*** 90.14*** 72.45*** 44.77*** 12.01*** 92.81*** 131.59*** 84.07*** 95.85*** 52.48*** 16.19*** 83.44***

(0.00%) (0.00%) (0.00%) (0.00%) (0.09%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.01%) (0.00%)SRBBCAT 53.79*** 43.25*** 46.08*** 21.10*** 5.80** 51.77*** 92.03*** 36.28*** 95.13*** 31.45*** 10.81*** 48.05***

(0.00%) (0.00%) (0.00%) (0.00%) (1.88%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.13%) (0.00%)SRGLU 82.68*** 96.89*** 54.49*** 17.40*** 19.70*** 36.71*** 102.99*** 92.78*** 65.56*** 35.10*** 18.95*** 68.47***

(0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%)SRUSWD 38.61*** 76.59*** 5.34** 35.43*** 2.88* 61.63*** 62.63*** 92.77*** 19.32*** 7.31*** 0.93 14.06***

(0.00%) (0.00%) (2.11%) (0.00%) (9.41%) (0.00%) (0.00%) (0.00%) (0.00%) (0.10%) (33.57%) (0.03%)

*Null hypothesis rejected at a level >90%, **Null hypothesis rejected a level >95%, ***Null hypothesis rejected at a level >99%.

Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: Swiss Re Global Cat Bond Index, SRBBCAT: Swiss Re BB Cat Bond Index,SRGLUCAT: Swiss Re Global Unhedged Cat Bond Index, SRUSWIND: Swiss Re US Wind Cat Bond Index. Asset Classes - USEQ:US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCI World ex-USA), EMEQ: Emerging Markets Equities (MSCIEmerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US Treasury Bonds (Bloomberg Barclays US Treasury), USCB:US Corporate Bonds (Bloomberg Barclays US Corporate Bonds), USHYB: US High-Yield Bonds (Bloomberg Barclays US High-Yield Bonds),COMM: Commodities (S & P Goldman Sachs Commodity). Factors - VALUE: Value Factor (Long: MSCI US Value - Short: MSCI US Growth),SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap), MOM: Momentum Factor (Long: MSCI US Momentum - Short:MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility - Short: MSCI US), MORTG: Mortgage Factor (Long: BarclaysUS MBS - Short: MSCI US), DEFAULT: Default Factor (Long: Bloomberg Barclays US Corporate Bonds - Short: Bloomberg Barclays USTreasury Bonds), TERM: Term Factor (Long: Bloomberg Barclays US Treasury 10-20y - Short: Bloomberg Barclays US Short Treasury), HY:High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short: Bloomberg Barclays US Corporate Bonds), C.Curve: CommodityCurve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P Goldman Sachs Commodity), EQ: Equity Market Factor (Long: MSCI US -Short: 1 Month T-Bill).

44

Table 9Statistics of the Sharpe ratio increases

This table reports the statistics of the dierence in Sharpe ratios with and without Cat Bonds (DSRt = SRa,t − SRs,t) for asset classes withand without short selling allowed (Panel A and B) and for factors with and without short selling allowed (Panel C and D). To obtain theportfolios' Sharpe ratios, the DCC estimates and the weekly average returns are used. US 1M T-Bill is the risk-free asset. The rst column

shows the average µ = 1T

∑Tt=1DSRt. The second column is the standard deviation σ =

√1T(DSRt − µ)2. The third is the median. The

fourth the minimum (Min). The fth the maximum (Max).

Panel A: Asset Classes

µ σ Median Min MaxSRCAT 0.6167*** 0.3140 0.6798 0.0005 1.2513

SRGLCAT 0.6454*** 0.3285 0.7075 0.0005 1.3362SRBBCAT 0.5890*** 0.3499 0.6312 0.0000 1.3312SRGLUCAT 0.3890*** 0.2026 0.4017 0.0005 0.9160SRUSWIND 0.4473*** 0.2352 0.4869 0.0000 0.8932

Panel B: Asset Classes - Long only

µ σ Median Min MaxSRCAT 0.6157*** 0.3098 0.6799 0.0005 1.1824

SRGLCAT 0.6428*** 0.3233 0.7006 0.0005 1.2525SRBBCAT 0.5898*** 0.3477 0.6302 0.0001 1.2577SRGLUCAT 0.3631*** 0.1969 0.3673 0.0000 0.7912SRUSWIND 0.4514*** 0.2337 0.4942 0.0000 0.8645

Panel C: Factors

µ σ Median Min MaxSRCAT 0.6178*** 0.3009 0.6719 0.0021 1.1984

SRGLCAT 0.6462*** 0.3132 0.6981 0.0023 1.2712SRBBCAT 0.6030*** 0.3259 0.6233 0.0002 1.2837SRGLUCAT 0.3883*** 0.1904 0.3870 0.0038 0.8525SRUSWIND 0.4424*** 0.2304 0.4860 0.0002 0.9028

Panel D: Factors - Long only

µ σ Median Min MaxSRCAT 0.6129*** 0.3002 0.6665 0.0019 1.1708

SRGLCAT 0.6395*** 0.3133 0.6946 0.0030 1.2405SRBBCAT 0.5872*** 0.3371 0.6160 0.0001 1.2533SRGLUCAT 0.3847*** 0.1902 0.3834 0.0048 0.8402SRUSWIND 0.4476*** 0.2266 0.4905 0.0000 0.8969

*Null hypothesis rejected at a level >90%, **Null hypothesis rejected a level >95%, ***Null hypothesis rejected at a level >99%.

Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: Swiss Re Global Cat Bond Index, SRBBCAT: Swiss Re BB Cat Bond Index,SRGLUCAT: Swiss Re Global Unhedged Cat Bond Index, SRUSWIND: Swiss Re US Wind Cat Bond Index. Asset Classes - USEQ:US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCI World ex-USA), EMEQ: Emerging Markets Equities (MSCIEmerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US Treasury Bonds (Bloomberg Barclays US Treasury), USCB:US Corporate Bonds (Bloomberg Barclays US Corporate Bonds), USHYB: US High-Yield Bonds (Bloomberg Barclays US High-Yield Bonds),COMM: Commodities (S & P Goldman Sachs Commodity). Factors - VALUE: Value Factor (Long: MSCI US Value - Short: MSCI US Growth),SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap), MOM: Momentum Factor (Long: MSCI US Momentum - Short:MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility - Short: MSCI US), MORTG: Mortgage Factor (Long: BarclaysUS MBS - Short: MSCI US), DEFAULT: Default Factor (Long: Bloomberg Barclays US Corporate Bonds - Short: Bloomberg Barclays USTreasury Bonds), TERM: Term Factor (Long: Bloomberg Barclays US Treasury 10-20y - Short: Bloomberg Barclays US Short Treasury), HY:High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short: Bloomberg Barclays US Corporate Bonds), C.Curve: CommodityCurve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P Goldman Sachs Commodity), EQ: Equity Market Factor (Long: MSCI US -Short: 1 Month T-Bill).

45

Table 10Statistics of the squared diversication ratio increases

This table reports the statistics of the dierence in squared diversication ratios (DDR2t = DR2

a,t −DR2s,t) for asset classes with and without

short selling allowed (Panel A and B) and for factors with and without short selling allowed (Panel C and D). To obtain the Sharpe ratio ofthe portfolios, the DCC estimates and the weekly average returns are used. The risk-free used is the US 1M T-Bill. The rst column shows

the average µ = 1T

∑Tt=1DSRt. The second column is the standard deviation σ =

√1T(DSRt − µ)2. The third is the median. The fourth the

minimum (Min). The fth the maximum (Max).

Panel A: Asset Classes

µ σ Median Min MaxSRCAT 1.1194*** 0.2106 1.1088 0.5003 2.3957

SRGLCAT 1.1223*** 0.2129 1.1089 0.5038 2.3941SRBBCAT 0.9785*** 0.2037 0.9923 0.3620 2.1076SRGLUCAT 0.5751*** 0.1850 0.5908 0.1071 1.4130SRUSWIND 0.9736*** 0.2250 0.9943 0.3801 1.5446

Panel B: Asset Classes - Long only

µ σ Median Min MaxSRCAT 0.9977*** 0.2294 0.9982 0.3336 2.1920

SRGLCAT 0.9944*** 0.2346 0.9968 0.3235 2.1856SRBBCAT 0.8733*** 0.2130 0.8903 0.2595 1.9300SRGLUCAT 0.4909*** 0.1915 0.5104 0.0732 1.2585SRUSWIND 0.9051*** 0.2205 0.9409 0.2784 1.4695

Panel C: Factors

µ σ Median Min MaxSRCAT 0.8985*** 0.2814 0.8712 0.2921 2.2827

SRGLCAT 0.8755*** 0.2820 0.8459 0.2848 2.2475SRBBCAT 0.9104*** 0.2583 0.9145 0.1924 1.9002SRGLUCAT 0.7631*** 0.2251 0.7344 0.2578 1.8098SRUSWIND 0.8004*** 0.2550 0.7701 0.2175 1.5980

Panel D: Factors - Long only

µ σ Median Min MaxSRCAT 0.8815*** 0.2783 0.8541 0.2921 2.2876

SRGLCAT 0.8584*** 0.2789 0.8267 0.2848 2.2524SRBBCAT 0.8970*** 0.2559 0.8995 0.1891 1.8756SRGLCAT 0.7522*** 0.2226 0.7188 0.2511 1.8098SRUSWIND 0.7998*** 0.2552 0.7688 0.2146 1.5980

*Null hypothesis rejected at a level >90%, **Null hypothesis rejected a level >95%, ***Null hypothesis rejected at a level >99%.

Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: Swiss Re Global Cat Bond Index, SRBBCAT: Swiss Re BB Cat Bond Index,SRGLUCAT: Swiss Re Global Unhedged Cat Bond Index, SRUSWIND: Swiss Re US Wind Cat Bond Index. Asset Classes - USEQ:US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCI World ex-USA), EMEQ: Emerging Markets Equities (MSCIEmerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US Treasury Bonds (Bloomberg Barclays US Treasury), USCB:US Corporate Bonds (Bloomberg Barclays US Corporate Bonds), USHYB: US High-Yield Bonds (Bloomberg Barclays US High-Yield Bonds),COMM: Commodities (S & P Goldman Sachs Commodity). Factors - VALUE: Value Factor (Long: MSCI US Value - Short: MSCI US Growth),SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap), MOM: Momentum Factor (Long: MSCI US Momentum - Short:MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility - Short: MSCI US), MORTG: Mortgage Factor (Long: BarclaysUS MBS - Short: MSCI US), DEFAULT: Default Factor (Long: Bloomberg Barclays US Corporate Bonds - Short: Bloomberg Barclays USTreasury Bonds), TERM: Term Factor (Long: Bloomberg Barclays US Treasury 10-20y - Short: Bloomberg Barclays US Short Treasury), HY:High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short: Bloomberg Barclays US Corporate Bonds), C.Curve: CommodityCurve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P Goldman Sachs Commodity), EQ: Equity Market Factor (Long: MSCI US -Short: 1 Month T-Bill).

46

Table 11Second-order stochastic dominance eciency tests

This table reports the statistics and p-values (in parentheses) of the SSDE test (second-order stochastic dominance (SD) eciency) (Scaillet andTopaloglou, 2010). The test is performed according to the two-step procedure of Daskalaki et al. (2017). The null hypothesis is that universeof asset classes or factors is SD ecient at the second-order compared to the same universe with Cat Bonds added. Panel A reports resultsfor the full sample (January 4, 2002 - June 30, 2017), business cycles from the NBER and sub-samples obtained from the turbulence index(Kritzman and Li, 2010). Panel B presents the test results for regimes determined from a Markov-Switching model (Hardy, 2001) on returnsof the Bloomber Barclays US Aggregate Bond index and for those from returns of the S&P500.

Panel A: Full sample, NBER and turbulence index

Full sample NBER Turbulence IndexNormal Recession Calm Turbulent

AssetClasses

Factors AssetClasses

Factors AssetClasses

Factors AssetClasses

Factors AssetClasses

Factors

SRCAT 0.2493 0.8299 0.1718 0.5852 0.2007 0.3004 0.2005 0.4368 0.2976 0.0894(67.0%) (53.0%) (88.3%) (53.0%) (87.0%) (86.0%) (63.7%) (56.3%) (63.7%) (60.7%)

SRGLCAT 0.2468 0.8528 0.1622 0.4386 0.1927 0.0530 0.1983 0.3607 0.2679 0.0884(67.0%) (52.0%) (90.7%) (60.3%) (90.7%) (91.0%) (62.3%) (61.3%) (62.3%) (72.0%)

SRBBCAT 0.2092 0.6461 0.1658 0.4691 0.1991 0.0542 0.1739 0.3434 0.2951 0.0775(74.7%) (53.3%) (84.3%) (63.0%) (84.3%) (89.3%) (67.3%) (59.7%) (67.3%) (66.3%)

SRGLUCAT 0.2403 0.7482 0.1707 0.6001 0.2034 0.0558 0.1992 0.4467 0.3022 0.0888(66.7%) (50.7%) (92.3%) (55.0%) (92.3%) (84.3%) (64.0%) (53.3%) (64.0%) (60.3%)

SRUSWIND 0.2292 0.6226 0.1658 0.5360 0.2116 0.0542 0.1973 0.3991 0.3148 0.0880(74.0%) (53.0%) (90.0%) (61.3%) (90.0%) (83.3%) (61.3%) (56.0%) (61.3%) (67.0%)

Panel B: Markov-Switching

Markov Switching - US aggregate bond Markov Switching - S&P500Low volatility High volatility Low volatility High volatility

AssetClasses

Factors AssetClasses

Factors AssetClasses

Factors Assetclasses

Factors

SRCAT 0.1837 0.5802 0.1572 0.0980 0.1325 0.2795 0.2372 0.0813(66.7%) (43.3%) (66.7%) (84.3%) (56.0%) (52.3%) (56.0%) (75.3%)

SRGLCAT 0.1658 0.4441 0.1487 0.0884 0.1279 0.2198 0.2315 0.0785(82.7%) (47.3%) (82.7%) (90.0%) (59.3%) (61.3%) (59.3%) (82.3%)

SRBBCAT 0.1665 0.4721 0.1425 0.0888 0.1155 0.2076 0.2361 0.0709(76.0%) (43.3%) (76.0%) (88.7%) (64.3%) (60.7%) (64.3%) (79.3%)

SRGLUCAT 0.1840 0.5979 0.1551 0.0982 0.1324 0.2871 0.2378 0.0812(68.7%) (44.0%) (68.7%) (86.0%) (63.0%) (53.0%) (63.0%) (74.0%)

SRUSWIND 0.1780 0.5396 0.1551 0.0950 0.1306 0.2619 0.2363 0.0801(73.0%) (41.3%) (73.0%) (88.3%) (62.7%) (57.7%) (62.7%) (67.7%)

*Null hypothesis rejected at a level >90%, **Null hypothesis rejected a level >95%, ***Null hypothesis rejected at a level >99%.

Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: Swiss Re Global Cat Bond Index, SRBBCAT: Swiss Re BB Cat Bond Index,SRGLUCAT: Swiss Re Global Unhedged Cat Bond Index, SRUSWIND: Swiss Re US Wind Cat Bond Index. Asset Classes - USEQ:US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCI World ex-USA), EMEQ: Emerging Markets Equities (MSCIEmerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US Treasury Bonds (Bloomberg Barclays US Treasury), USCB:US Corporate Bonds (Bloomberg Barclays US Corporate Bonds), USHYB: US High-Yield Bonds (Bloomberg Barclays US High-Yield Bonds),COMM: Commodities (S & P Goldman Sachs Commodity). Factors - VALUE: Value Factor (Long: MSCI US Value - Short: MSCI US Growth),SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap), MOM: Momentum Factor (Long: MSCI US Momentum - Short:MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility - Short: MSCI US), MORTG: Mortgage Factor (Long: BarclaysUS MBS - Short: MSCI US), DEFAULT: Default Factor (Long: Bloomberg Barclays US Corporate Bonds - Short: Bloomberg Barclays USTreasury Bonds), TERM: Term Factor (Long: Bloomberg Barclays US Treasury 10-20y - Short: Bloomberg Barclays US Short Treasury), HY:High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short: Bloomberg Barclays US Corporate Bonds), C.Curve: CommodityCurve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P Goldman Sachs Commodity), EQ: Equity Market Factor (Long: MSCI US -Short: 1 Month T-Bill).

47

Table 12Out-of-sample statistics - Maximum Sharpe ratio

This table reports the performance measures for the maximum Sharpe ratio portfolios. The portfolios returns and weights are calculatedout-of-sample using a rolling-window technique of length K=120. The benchmark portfolio is the one constructed from asset classes with shortselling (Panel A) and without short selling (Panel B) or factors with short selling (Panel C) and without short selling (Panel D). The othercolumns are portfolios constructed using the benchmark universe plus a Cat Bonds index. The performance measures used are the Sharpe ratio(SR), the portfolio turnover (TRR) and the conditional Sharpe ratio (CSR). The p-value corresponds to the test from Memmel (2003) wherethe null hypothesis is that the Sharpe ratio of the benchmark portfolio is equal to that of the portfolio with Cat Bonds included.

Panel A: Asset Classes

SRCAT SRGLCAT SRBBCAT SRGLUCAT SRUSWIND BenchmarkSR 0.2237** 0.2259** 0.1595 0.2250** 0.1879 0.1776

p-value 3.2% 2.7% 79.9% 2.4% 36.4%TR 25.2% 24.4% 29.6% 25.7% 24.9% 37.8%CSR 0.0384 0.0374 0.0297 0.0396 0.0202 0.0492

Panel B: Asset Classes - Long only

SRCAT SRGLCAT SRBBCAT SRGLUCAT SRUSWIND BenchmarkSR 0.1987*** 0.2014*** 0.1332** 0.1887*** 0.1585** 0.0968

p-value 0.0% 0.0% 4.9% 0.1% 3.2%TR 4.9% 4.7% 6.0% 5.3% 5.8% 9.8%CSR 0.0219 0.0211 0.0189 0.0215 0.0155 0.0160

Panel C: Factors

SRCAT SRGLCAT SRBBCAT SRGLUCAT SRUSWIND BenchmarkSR 0.3216*** 0.3367*** 0.2270*** 0.3188*** 0.1962** 0.1091

p-value 0.0% 0.0% 0.1% 0.0% 1.9%TR 17.8% 17.1% 18.8% 18.4% 22.3% 36.2%CSR 0.0699 0.0756 0.0394 0.0794 0.0122 0.0221

Panel D: Factors - Long only

SRCAT SRGLCAT SRBBCAT SRGLUCAT SRUSWIND BenchmarkSR 0.2888*** 0.3003*** 0.1999*** 0.2831*** 0.1812* 0.1161

p-value 0.0% 0.0% 0.7% 0.0% 5.8%TR 6.1% 5.9% 6.6% 6.1% 7.4% 10.6%CSR 0.0471 0.0492 0.0291 0.0522 0.0121 0.0186

*Null hypothesis rejected at a level >90%, **Null hypothesis rejected a level >95%, ***Null hypothesis rejected at a level >99%.

The benchmark portfolio is simply the portfolio of the universe of asset classes (Panel A and B) or factors (Panel C and D) whose weights areobtained at each period by the maximization of the Sharpe ratio. Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: Swiss Re GlobalCat Bond Index, SRBBCAT: Swiss Re BB Cat Bond Index, SRGLUCAT: Swiss Re Global Unhedged Cat Bond Index, SRUSWIND: Swiss ReUS Wind Cat Bond Index. Asset Classes - USEQ: US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCI Worldex-USA), EMEQ: Emerging Markets Equities (MSCI Emerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US TreasuryBonds (Bloomberg Barclays US Treasury), USCB: US Corporate Bonds (Bloomberg Barclays US Corporate Bonds), USHYB: US High-YieldBonds (Bloomberg Barclays US High-Yield Bonds), COMM: Commodities (S & P Goldman Sachs Commodity). Factors - VALUE: ValueFactor (Long: MSCI US Value - Short: MSCI US Growth), SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap),MOM: Momentum Factor (Long: MSCI US Momentum - Short: MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility -Short: MSCI US), MORTG: Mortgage Factor (Long: Barclays US MBS - Short: MSCI US), DEFAULT: Default Factor (Long: BloombergBarclays US Corporate Bonds - Short: Bloomberg Barclays US Treasury Bonds), TERM: Term Factor (Long: Bloomberg Barclays US Treasury10-20y - Short: Bloomberg Barclays US Short Treasury), HY: High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short:Bloomberg Barclays US Corporate Bonds), C.Curve: Commodity Curve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P GoldmanSachs Commodity), EQ: Equity Market Factor (Long: MSCI US - Short: 1 Month T-Bill).

48

Table 13Out-of-sample statistics - Maximum diversication ratio

This table reports the performance measures for the maximum diversication portfolios. The returns and weights of the portfolios are calculatedout-of-sample using a rolling-window technique of length K=120. The benchmark portfolio is the one constructed from asset classes with shortselling (Panel A) and without short selling (Panel B) or factors with short selling (Panel C) and without short selling allowed (Panel D) . Theother columns are portfolios constructed using the benchmark universe plus a Cat Bonds index. The performance measures used are the Sharperatio (SR), the portfolio turnover (TRR) and the conditional Sharpe ratio (CSR). The p-value corresponds to the test from Memmel (2003)where the null hypothesis is that the Sharpe ratio of the benchmark portfolio is equal to the one of the portfolio with Cat Bonds.

Panel A: Asset Classes

SRCAT SRGLCAT SRBBCAT SRGLUCAT SRUSWIND BenchmarkSR 0.2258*** 0.2268*** 0.1975*** 0.1917*** 0.1762** 0.1299

p-value 0.0% 0.0% 0.0% 0.0% 1.6%TR 6.6% 6.6% 6.6% 7.1% 6.8% 6.7%CSR 0.0465 0.0440 0.0370 0.0392 0.0238 0.0254

Panel B: Asset Classes - Long only

SRCAT SRGLCAT SRBBCAT SRGLUCAT SRUSWIND BenchmarkSR 0.1752*** 0.1762*** 0.1423*** 0.1430*** 0.1383*** 0.0866

p-value 0.0% 0.0% 0.1% 0.0% 0.8%TR 2.5% 2.6% 2.5% 2.5% 2.5% 2.0%CSR 0.0353 0.0339 0.0258 0.0261 0.0188 0.0173

Panel C: Factors

SRCAT SRGLCAT SRBBCAT SRGLUCAT SRUSWIND BenchmarkSR 0.1713*** 0.1709*** 0.1529*** 0.1585*** 0.1525*** 0.1097

p-value 0.0% 0.0% 0.0% 0.0% 0.2%TR 8.0% 8.0% 7.9% 8.3% 8.1% 7.6%CSR 0.0340 0.0326 0.0281 0.0307 0.0272 0.0213

Panel D: Factors - Long only

SRCAT SRGLCAT SRBBCAT SRGLUCAT SRUSWIND BenchmarkSR 0.1317*** 0.1305*** 0.1178*** 0.1188*** 0.1066* 0.0793

p-value 0.0% 0.0% 0.3% 0.0% 5.2%TR 3.8% 3.9% 3.8% 3.9% 3.7% 3.5%CSR 0.0227 0.0212 0.0187 0.0202 0.0161 0.0151

*Null hypothesis rejected at a level >90%, **Null hypothesis rejected a level >95%, ***Null hypothesis rejected at a level >99%.

The benchmark portfolio is simply the portfolio of the universe of asset classes (Panel A and B) or factors (Panel C and D) whose weights areobtained at each period by the maximization of the diversication ratio. Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: SwissRe Global Cat Bond Index, SRBBCAT: Swiss Re BB Cat Bond Index, SRGLUCAT: Swiss Re Global Unhedged Cat Bond Index, SRUSWIND:Swiss Re US Wind Cat Bond Index. Asset Classes - USEQ: US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCIWorld ex-USA), EMEQ: Emerging Markets Equities (MSCI Emerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US TreasuryBonds (Bloomberg Barclays US Treasury), USCB: US Corporate Bonds (Bloomberg Barclays US Corporate Bonds), USHYB: US High-YieldBonds (Bloomberg Barclays US High-Yield Bonds), COMM: Commodities (S & P Goldman Sachs Commodity). Factors - VALUE: ValueFactor (Long: MSCI US Value - Short: MSCI US Growth), SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap),MOM: Momentum Factor (Long: MSCI US Momentum - Short: MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility -Short: MSCI US), MORTG: Mortgage Factor (Long: Barclays US MBS - Short: MSCI US), DEFAULT: Default Factor (Long: BloombergBarclays US Corporate Bonds - Short: Bloomberg Barclays US Treasury Bonds), TERM: Term Factor (Long: Bloomberg Barclays US Treasury10-20y - Short: Bloomberg Barclays US Short Treasury), HY: High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short:Bloomberg Barclays US Corporate Bonds), C.Curve: Commodity Curve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P GoldmanSachs Commodity), EQ: Equity Market Factor (Long: MSCI US - Short: 1 Month T-Bill).

49

Table 14Out-of-sample statistics - SSDE (second-order stochastic dominance eciency)

This table reports the performance measures for the SSDE portfolios obtained with the two-step procedure of Daskalaki et al. (2017). Thereturns and weights of the portfolios are calculated out-of-sample using a rolling-window technique of length K=120. The benchmark portfoliois the one constructed from asset classes with short selling (Panel A) and without short selling (Panel B) or factors with short selling (PanelC) and without short selling (Panel D). The other columns are portfolios constructed using the benchmark universe plus a Cat Bonds index.The performance measures used are the Sharpe ratio (SR), the portfolio turnover (TRR) and the conditional Sharpe ratio (CSR). The p-valuecorresponds to the test from Memmel (2003) where the null hypothesis is that the Sharpe ratio of the benchmark portfolio is equal to the oneof the portfolio with Cat Bonds.

Panel A: Asset Classes

SRCAT SRGLCAT SRBBCAT SRGLUCAT SRUSWIND BenchmarkSR 0.1206* 0.1181* 0.0898 0.1136 0.1045 0.0886

p-value 9.0% 8.9% 47.5% 14.3% 25.0%TR 557.9% 562.4% 610.0% 562.7% 601.5% 507.5%CSR 0.0135 0.0132 0.0106 0.0139 0.0112 0.0183

Panel B: Asset Classes - Long only

SRCAT SRGLCAT SRBBCAT SRGLUCAT SRUSWIND BenchmarkSR 0.3213*** 0.3368*** 0.2195** 0.2970*** 0.1539 0.1153

p-value 0.0% 0.0% 1.0% 0.0% 18.6%TR 2.2% 1.9% 2.9% 2.8% 6.8% 4.7%MSR 0.0308 0.0322 0.0177 0.0342 0.0060 0.0223

Panel C: Factors

SRCAT SRGLCAT SRBBCAT SRGLUCAT SRUSWIND BenchmarkSR 0.1969*** 0.2150*** 0.1298* 0.1994*** 0.1542** 0.0710

p-value 0.1% 0.0% 7.4% 0.1% 2.0%TR 1223.7% 1254.1% 1214.9% 3730.3% 975.2% 692.5%CSR 0.0286 0.0306 0.0183 0.0271 0.0222 0.0083

Panel D: Factors - Long only

SRCAT SRGLCAT SRBBCAT SRGLUCAT SRUSWIND BenchmarkSR 0.2742*** 0.2918*** 0.1724*** 0.2431*** 0.1361** 0.0319

p-value 0.0% 0.0% 0.1% 0.0% 1.3%TR 6.6% 6.1% 6.4% 5.7% 6.8% 7.5%CSR 0.0202 0.0224 0.0103 0.0194 0.0221 0.0036

*Null hypothesis rejected at a level >90%, **Null hypothesis rejected a level >95%, ***Null hypothesis rejected at a level >99%.

The benchmark portfolio is simply the portfolio of the universe of asset classes (Panel A and B) or factors (Panel C and D) whose weights areobtained at each period by the maximization of the SSDE criterion. Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: Swiss ReGlobal Cat Bond Index, SRBBCAT: Swiss Re BB Cat Bond Index, SRGLUCAT: Swiss Re Global Unhedged Cat Bond Index, SRUSWIND:Swiss Re US Wind Cat Bond Index. Asset Classes - USEQ: US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCIWorld ex-USA), EMEQ: Emerging Markets Equities (MSCI Emerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US TreasuryBonds (Bloomberg Barclays US Treasury), USCB: US Corporate Bonds (Bloomberg Barclays US Corporate Bonds), USHYB: US High-YieldBonds (Bloomberg Barclays US High-Yield Bonds), COMM: Commodities (S & P Goldman Sachs Commodity). Factors - VALUE: ValueFactor (Long: MSCI US Value - Short: MSCI US Growth), SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap),MOM: Momentum Factor (Long: MSCI US Momentum - Short: MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility -Short: MSCI US), MORTG: Mortgage Factor (Long: Barclays US MBS - Short: MSCI US), DEFAULT: Default Factor (Long: BloombergBarclays US Corporate Bonds - Short: Bloomberg Barclays US Treasury Bonds), TERM: Term Factor (Long: Bloomberg Barclays US Treasury10-20y - Short: Bloomberg Barclays US Short Treasury), HY: High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short:Bloomberg Barclays US Corporate Bonds), C.Curve: Commodity Curve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P GoldmanSachs Commodity), EQ: Equity Market Factor (Long: MSCI US - Short: 1 Month T-Bill).

50

Figure 1Turbulence index

The gure presents the evolution of the Mahalanobis distance dt = (yt−µ)Σ−1(yt−µ)′. The horizontal line represents the threshold betweenturbulent (top) and quiet (bottom) observations.

51

Figure 2Regime Switching

The gure shows the most probable regime series estimated by means of the Viterbi (1967) algorithm from the Regime Switching models onthe S&P 500 index (top) and the Bloomberg Barclays US Aggregate Bond index (bottom).

52

Figure 3NBER

The gure shows the regimes obtained from the NBER data. 1 represents a recession and 0 a so-called normal period.

53

Figure 4Stochastic dominance

The gure shows examples of rst-order stochastic dominance (top) (GA(x) ≤ FB(x)) and second-order stochastic dominance (bottom)(∫∞∞ [GA(x)− FB(x)]dx ≤ 0

). F is the probability cumulative function.

54

Figure 5DCC (Dynamic Conditional Correlation) estimates - Asset Classes

This gure presents the estimates of the DCC (Dynamic Conditional Correlation) model (Engle, 2002) on the asset classes and Cat Bonds (we present only the Swiss Re Cat Bond index). CatBonds - SRCAT: Swiss Re Cat Bond Index. Asset Classes - USEQ: US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCI World ex-USA), EMEQ: Emerging MarketsEquities (MSCI Emerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US Treasury Bonds (Bloomberg Barclays US Treasury), USCB: US Corporate Bonds (BloombergBarclays US Corporate Bonds), USHYB: US High-Yield Bonds (Bloomberg Barclays US High-Yield Bonds), COMM: Commodities (S & P Goldman Sachs Commodity).

55

Figure 6DCC (Dynamic Conditional Correlation) estimates - Factors

This gure presents the estimates of the DCC (Dynamic Conditional Correlation) model (Engle, 2002) on the factors and Cat Bonds (we present only the Swiss Re Cat Bond index). CatBonds - SRCAT: Swiss Re Cat Bond Index. Factors - VALUE: Value Factor (Long: MSCI US Value - Short: MSCI US Growth), SIZE: Size Factor (Long: MSCI US Small Cap - Short:MSCI US Large Cap), MOM: Momentum Factor (Long: MSCI US Momentum - Short: MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility - Short: MSCI US), MORTG:Mortgage Factor (Long: Barclays US MBS - Short: MSCI US), DEFAULT: Default Factor (Long: Bloomberg Barclays US Corporate Bonds - Short: Bloomberg Barclays US Treasury Bonds),TERM: Term Factor (Long: Bloomberg Barclays US Treasury 10-20y - Short: Bloomberg Barclays US Short Treasury), HY: High-Yield Factor (Long: Bloomberg Barclays US High-YieldBonds - Short: Bloomberg Barclays US Corporate Bonds), C.Curve: Commodity Curve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P Goldman Sachs Commodity), EQ: EquityMarket Factor (Long: MSCI US - Short: 1 Month T-Bill).

56

Figure 7Increase in the Sharpe ratio of the tangent portfolio due to the addition of Cat Bonds

This gure shows the time series of the increase in the Sharpe ratio of the tangent portfolio with Cat Bonds added. The tangent portfoliois rst constructed without Cat Bonds, then with Cat Bonds, and the dierence between the ratios is calculated. The correlations used arefrom the DCC model (Engle, 2002). The chart at the top left is for the case with the benchmark portfolio constructed from asset classeswithout any constraint. The chart at the top right is related to the asset class framework without allowing short selling. The bottom leftpertains the unrestricted factors framework. The bottom right concerns also the context of factors but when short selling is not allowed.The benchmark portfolio is the portfolio constructed from asset classes (top) or factors (bottom) whose weights are obtained by maximizingthe Sharpe ratio. Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: Swiss Re Global Cat Bond Index, SRBBCAT: Swiss ReBB Car Bond Index, SRGLUCAT: Swiss Re Global Unhedged Cat Bond Index, SRUSWIND: Swiss Re US Wind Cat Bond Index. AssetClasses - USEQ: US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCI World ex-USA), EMEQ: Emerging MarketsEquities (MSCI Emerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US Treasury Bonds (Bloomberg Barclays US Treasury),USCB: US Corporate Bonds (Bloomberg Barclays US Corporate Bonds), USHYB: US High-Yield Bonds (Bloomberg Barclays US High-YieldBonds), COMM: Commodities (S & P Goldman Sachs Commodity). Factors - VALUE: Value Factor (Long: MSCI US Value - Short: MSCIUS Growth), SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap), MOM: Momentum Factor (Long: MSCI USMomentum - Short: MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility - Short: MSCI US), MORTG: Mortgage Factor(Long: Barclays US MBS - Short: MSCI US), DEFAULT: Default Factor (Long: Bloomberg Barclays US Corporate Bonds - Short: BloombergBarclays US Treasury Bonds), TERM: Term Factor (Long: Bloomberg Barclays US Treasury 10-20y - Short: Bloomberg Barclays US ShortTreasury), HY: High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short: Bloomberg Barclays US Corporate Bonds),C.Curve: Commodity Curve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P Goldman Sachs Commodity), EQ: Equity Market Factor(Long: MSCI US - Short: 1 Month T-Bill).

57

Figure 8Guy Carpenter U.S. Property Rate on Line Index

This gure shows the index of the global reinsurance rate movements that has been maintained by Guy Carpenter since 1990. It covers allmajor catastrophe reinsurance markets.

58

Figure 9Increases in the squared diversication ratios of the maximum diversication portfolio due to the addition of Cat Bonds

This gure shows the time series of the increases in the squared diversication ratio of the maximum diversication portfolio with Cat Bondsadded. The portfolio is rst constructed without Cat Bonds, then with Cat Bonds, and the dierence between the ratios is calculated. Thecorrelations used are from the DCC model (Engle, 2002). The chart at the top left is for the case where the benchmark portfolio is constructedfrom asset classes without any constraint. The chart at the top right pertains the asset class framework without allowing short selling. Thebottom left concerns the unrestricted factors framework. The bottom right is also in the context of factors but when short selling is not allowed.The benchmark portfolio is the portfolio constructed from asset classes (top) or factors (bottom) whose weight are obtained by maximizingthe diversication ratio. Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: Swiss Re Global Cat Bond Index, SRBBCAT: SwissRe BB Cat Bond Index, SRGLUCAT: Swiss Re Global Unhedged Cat Bond Index, SRUSWIND: Swiss Re US Wind Cat Bond Index. AssetClasses - USEQ: US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCI World ex-USA), EMEQ: Emerging MarketsEquities (MSCI Emerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US Treasury Bonds (Bloomberg Barclays US Treasury),USCB: US Corporate Bonds (Bloomberg Barclays US Corporate Bonds), USHYB: US High-Yield Bonds (Bloomberg Barclays US High-YieldBonds), COMM: Commodities (S & P Goldman Sachs Commodity). Factors - VALUE: Value Factor (Long: MSCI US Value - Short: MSCIUS Growth), SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap), MOM: Momentum Factor (Long: MSCI USMomentum - Short: MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility - Short: MSCI US), MORTG: Mortgage Factor(Long: Barclays US MBS - Short: MSCI US), DEFAULT: Default Factor (Long: Bloomberg Barclays US Corporate Bonds - Short: BloombergBarclays US Treasury Bonds), TERM: Term Factor (Long: Bloomberg Barclays US Treasury 10-20y - Short: Bloomberg Barclays US ShortTreasury), HY: High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short: Bloomberg Barclays US Corporate Bonds),C.Curve: Commodity Curve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P Goldman Sachs Commodity), EQ: Equity Market Factor(Long: MSCI US - Short: 1 Month T-Bill).

59

Figure 10Omega ratio - Maximum Sharpe ratio

This gure shows the graphs of the Omega ratio (Keating and Shadwick, 2002) of maximum Sharpe ratio portfolios at dierent thresholds. Oneach graph, we present the curve for the benchmark portfolio (top left quadrant: asset classes with short selling, quadrant top right: asset classeswithout short selling, bottom left: factors with short selling, bottom right: factors without short selling) and where Cat Bonds are added to theinvestment universe. For each quadrant, the ve graphs show the addition of Cat Bonds according to the dierent proxies (Swiss Re Cat BondIndex, Swiss Re Global Cat Bond Index, Swiss Re BB Cat Bond Index, Swiss Re Global Unhedged Bond Index, Swiss Re US Wind Cat BondIndex). The scale is logarithmic. The Bench portfolio is simply the portfolio of the universe of asset classes (Top) or factors (Bottom) whoseweights are obtained each period by maximizing the Sharpe ratio. Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: Swiss Re GlobalCat Bond Index, SRBBCAT: Swiss Re BB Cat Bond Index, SRGLUCAT: Swiss Re Global Unhedged Cat Bond Index, SRUSWIND: Swiss ReUS Wind Cat Bond Index. Asset Classes - USEQ: US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCI Worldex-USA), EMEQ: Emerging Markets Equities (MSCI Emerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US TreasuryBonds (Bloomberg Barclays US Treasury), USCB: US Corporate Bonds (Bloomberg Barclays US Corporate Bonds), USHYB: US High-YieldBonds (Bloomberg Barclays US High-Yield Bonds), COMM: Commodities (S & P Goldman Sachs Commodity). Factors - VALUE: ValueFactor (Long: MSCI US Value - Short: MSCI US Growth), SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap),MOM: Momentum Factor (Long: MSCI US Momentum - Short: MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility -Short: MSCI US), MORTG: Mortgage Factor (Long: Barclays US MBS - Short: MSCI US), DEFAULT: Default Factor (Long: BloombergBarclays US Corporate Bonds - Short: Bloomberg Barclays US Treasury Bonds), TERM: Term Factor (Long: Bloomberg Barclays US Treasury10-20y - Short: Bloomberg Barclays US Short Treasury), HY: High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short:Bloomberg Barclays US Corporate Bonds), C.Curve: Commodity Curve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P GoldmanSachs Commodity), EQ: Equity Market Factor (Long: MSCI US - Short: 1 Month T-Bill).

60

Figure 11Omega ratio - Maximum diversication ratio

This gure shows the graphs of the Omega ratio (Keating and Shadwick, 2002) of maximum diversication portfolios at dierent thresholds. Oneach graph, we present the curve for the benchmark portfolio (top left quadrant: asset classes with short selling, quadrant top right: asset classeswithout short selling, bottom left: factors with short selling, bottom right: factors without short selling) and where Cat Bonds are added to theinvestment universe. For each quadrant, the ve graphs show the addition of Cat Bonds according to the dierent proxies (Swiss Re Cat BondIndex, Swiss Re Global Cat Bond Index, Swiss Re BB Cat Bond Index, Swiss Re Global Unhedged Bond Index, Swiss Re US Wind Cat BondIndex). The scale is logarithmic. The Bench portfolio is simply the portfolio of the universe of asset classes (Top) or factors (Bottom) whoseweights are obtained each period by maximizing the diversication ratio. Cat Bonds - SRCAT: Swiss Re Cat Bond Index SRGLCAT: SwissRe Global Cat Bond Index, SRBBCAT: Swiss Re BB Car Bond Index, SRGLUCAT: Swiss Re Global Unhedged Cat Bond Index, SRUSWIND:Swiss Re US Wind Cat Bond Index. Asset Classes - USEQ: US Equities (US - Datastream Market), XUSEQ: Global Equities ex-US (MSCIWorld ex-USA), EMEQ: Emerging Markets Equities (MSCI Emerging Markets), RE: Real Estate (US - Datastream REITs), USTB: US TreasuryBonds (Bloomberg Barclays US Treasury), USCB: US Corporate Bonds (Bloomberg Barclays US Corporate Bonds), USHYB: US High-YieldBonds (Bloomberg Barclays US High-Yield Bonds), COMM: Commodities (S & P Goldman Sachs Commodity). Factors - VALUE: ValueFactor (Long: MSCI US Value - Short: MSCI US Growth), SIZE: Size Factor (Long: MSCI US Small Cap - Short: MSCI US Large Cap),MOM: Momentum Factor (Long: MSCI US Momentum - Short: MSCI US), VOL: Factor Volatility (Long: MSCI US Minimum Volatility -Short: MSCI US), MORTG: Mortgage Factor (Long: Barclays US MBS - Short: MSCI US), DEFAULT: Default Factor (Long: BloombergBarclays US Corporate Bonds - Short: Bloomberg Barclays US Treasury Bonds), TERM: Term Factor (Long: Bloomberg Barclays US Treasury10-20y - Short: Bloomberg Barclays US Short Treasury), HY: High-Yield Factor (Long: Bloomberg Barclays US High-Yield Bonds - Short:Bloomberg Barclays US Corporate Bonds), C.Curve: Commodity Curve Factor (Long: S & P GSCI Dynamic Roll - Short: S & P GoldmanSachs Commodity), EQ: Equity Market Factor (Long: MSCI US - Short: 1 Month T-Bill).

61

Figure 12Omega ratio - SSDE (second-order stochastic dominance eciency)

This gure shows the graphs of the Omega ratio (Keating and Shadwick, 2002) of SSDE portfolios obtained by the two-step procedure ofDaskalaki et al. (2017) at dierent thresholds. On each graph, we present the curve for the benchmark portfolio (top left quadrant: asset classeswith short selling, quadrant top right: asset classes without short selling, bottom left: factors with short selling, bottom right: factors withoutshort selling) and where Cat Bonds are added to the investment universe. For each quadrant, the ve graphs show the addition of Cat Bondsaccording to the dierent proxies (Swiss Re Cat Bond Index, Swiss Re Global Cat Bond Index, Swiss Re BB Cat Bond Index, Swiss Re GlobalUnhedged Bond Index, Swiss Re US Wind Cat Bond Index). The scale is logarithmic. The Bench portfolio is simply the portfolio of the universeof asset classes (Top) or factors (Bottom) whose weights are obtained each period by maximizing the SSDE criterion. Cat Bonds - SRCAT:Swiss Re Cat Bond Index SRGLCAT: Swiss Re Global Cat Bond Index, SRBBCAT: Swiss Re BB Car Bond Index, SRGLUCAT: Swiss ReGlobal Unhedged Cat Bond Index, SRUSWIND: Swiss Re US Wind Cat Bond Index. Asset Classes - USEQ: US Equities (US - DatastreamMarket), XUSEQ: Global Equities ex-US (MSCI World ex-USA), EMEQ: Emerging Markets Equities (MSCI Emerging Markets), RE: RealEstate (US - Datastream REITs), USTB: US Treasury Bonds (Bloomberg Barclays US Treasury), USCB: US Corporate Bonds (BloombergBarclays US Corporate Bonds), USHYB: US High-Yield Bonds (Bloomberg Barclays US High-Yield Bonds), COMM: Commodities (S & PGoldman Sachs Commodity). Factors - VALUE: Value Factor (Long: MSCI US Value - Short: MSCI US Growth), SIZE: Size Factor (Long:MSCI US Small Cap - Short: MSCI US Large Cap), MOM: Momentum Factor (Long: MSCI US Momentum - Short: MSCI US), VOL: FactorVolatility (Long: MSCI US Minimum Volatility - Short: MSCI US), MORTG: Mortgage Factor (Long: Barclays US MBS - Short: MSCIUS), DEFAULT: Default Factor (Long: Bloomberg Barclays US Corporate Bonds - Short: Bloomberg Barclays US Treasury Bonds), TERM:Term Factor (Long: Bloomberg Barclays US Treasury 10-20y - Short: Bloomberg Barclays US Short Treasury), HY: High-Yield Factor (Long:Bloomberg Barclays US High-Yield Bonds - Short: Bloomberg Barclays US Corporate Bonds), C.Curve: Commodity Curve Factor (Long: S &P GSCI Dynamic Roll - Short: S & P Goldman Sachs Commodity), EQ: Equity Market Factor (Long: MSCI US - Short: 1 Month T-Bill).

.

62