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## Abstract

We study how traditional reinsurance and CAT bonds can be combined to build an optimal catastrophe insurance program. We develop a contingent claims model to investigate the imperfections and limitations of the reinsurance market stemming from financial distress costs and default risk. We find that the pricing markup and credit risk is typically larger for reinsurance contracts that cover the higher and less probable layers of losses. We show that the optimal hedging strategy is to cover small losses using reinsurance and to hedge higher losses by issuing a CAT bond. Our results demonstrate that this strategy significantly lowers the insurer’s cost of protection, expands his underwriting capacity, and yields higher shareholder values.

Increases in the frequency and severity of natural catastrophes in recent years have propelled the use of alternative risk transfer (ART) instruments for managing catastrophic risks. The outstanding catastrophe bond and insurance-linked securities market at the end of the second quarter of 2017 reached a new record of $29.3 billion.^{1} The catastrophe bond (CAT bond), the most important type of ART instrument in terms of volume, constitutes an interesting complement to traditional reinsurance for a catastrophe insurance firm seeking to expand its risk-bearing capacity.

CAT bonds, which are traded in capital markets, not only exhibit different characteristics than traditional reinsurance but also offer some advantages over it. For instance, CAT bonds are usually fully collateralized, and their credit risk is therefore small, especially in the case of the new hybrids launched since the subprime crisis, which are protected against the credit risk of the total return swap counterparty.^{2} In contrast, a reinsurer is inherently not without default risk, which can substantially reduce the insuring/hedging effectiveness of reinsurance. Reinsurance firms usually have pre-existing risk exposures, so their ability to take on additional catastrophe risks might be limited. It is documented empirically that reinsurers charge very high premiums for protection against high layers of losses that have low occurrence probabilities (see Exhibit 9 in Froot [2001]). Meanwhile, reinsurance presents advantages over ART instruments. For instance, traditional reinsurance offers potentially mutual benefits in business relationships, whereas a CAT bond sponsor receives no benefit from long-term business relationships with investors. Moreover, the reinsurer can monitor the insurer much more effectively than CAT bond investors can, meaning that the reinsurer should be less exposed to moral hazard, which could reduce the cost of protection. Later in the paper, we show that the differences between reinsurance and CAT bonds make these instruments complement each other.

There is a literature studying the interactions between ART instruments and reinsurance. Doherty and Richter [2002] use a mean-variance framework to show the benefits of combining indemnity-based gap reinsurance with an index hedge. Nell and Richter [2004], by way of an expected utility approach, underscore the substitution effects between traditional reinsurance and parametric CAT bonds for large losses. Their rationale for the high cost of catastrophe reinsurance is the reinsurers’ risk aversion for large losses. Lee and Yu [2007] use a contingent claims framework to study how the value of a reinsurer’s contract can be increased by issuing CAT bonds. Cummins and Song [2008] use a mean-variance framework to show that traditional reinsurance and derivative hedging display a substitution effect when the insurer assets and liabilities do not exhibit natural hedging characteristics. Cummins and Trainar [2009] argue that reinsurance should be preferred over securitization for relatively small and uncorrelated risks, but that securitization becomes more advantageous as the magnitude of potential losses and the correlation of risks begin to increase. Cummins and Weiss [2009] explain how the convergence of financial and (re)insurance markets has been driven by the increase in the frequency and severity of catastrophe risk, and argue that securitization expands (re)insurers’ risk-bearing capacity. Härdle and Cabrera [2010] perform the calibration of a parametric CAT bond that was sponsored by the Mexican government, and find that combining reinsurance with this CAT bond is optimal in the sense that it provides coverage for a lower cost and lower exposure at default than reinsurance itself.

Lakdawalla and Zanjani [2012] show that CAT bonds have important uses when buyers and reinsurers cannot contract over the division of assets in the event of insolvency. Using numerical simulations, they illustrate how CAT bonds improve efficiency in markets with correlated risks, or with uneven exposure of the insured parties to reinsurer default. Gibson et al. [2014] argue that the choice between traditional reinsurance and CAT bonds depends crucially on the information acquisition cost structure and on the degree of redundancy in the information produced. Hagendorff et al. [2014] document empirically that insurance firms that issue CAT bonds tend to have less risky underwriting portfolios with less exposure to catastrophe risks, and overall less need to hedge catastrophe risk. They also find that firms issuing CAT bonds tend to experience a reduction in their default risk and to increase their catastrophe exposure following a CAT bond issuance.

Cummins and Trainar [2009] point out that securitization is likely to play an important role in facilitating (re)insurers to shift optimal combinations of risk to the capital markets, especially catastrophic risks. However, theoretical models studying how to mix traditional reinsurance and ART instruments in an optimal way are scarce. Haslip and Kaishev [2010] develop a methodology for pricing reinsurance contracts consistent with the observed market prices of catastrophe bonds on the same underlying risk process but they do not study the combination of reinsurance and CAT bonds. Chang et al. [2015] present a model in which the insurer sets its optimal allocation between traditional reinsurance and a CAT bond by minimizing the total hedging cost, taking into account pricing markups (or loadings) for both instruments. The pricing markups are, however, treated as exogenous constants that are independent of the contracts’ characteristics. The catastrophe loss process and the total external hedging needs of the insurer are also fixed exogenously.

In this article, we develop a model in which these parameters are determined endogenously. The pricing markups of the hedging instruments are derived from market imperfections, and the loss process and hedging needs of the insurer are determined by way of a shareholder value maximization setup.^{3} Our framework complements Nell and Richter’s [2004] optimal allocation model, in which the explanation for the insurer’s demand for hedging is its risk aversion.

Corporate demand for hedging can be motivated by the desire to reduce the expected costs of financial distress (see e.g., Mayers and Smith [1982]), a key feature that we explicitly model in our paper. Nell and Richter [2004] model a CAT bond with a parametric trigger; however, since over 50% of risk capital issued during 2015 opted for indemnity triggers, we employ this type of trigger in our model. Further, unlike Nell and Richter [2004], we consider both the risk of moral hazard and the reinsurer’s risk of default.

This article studies how traditional reinsurance and CAT bonds can be combined to build an optimal catastrophe insurance program. Toward this end, we develop a contingent claims model that incorporates the reinsurer risk of insolvency, as well as the impact of financial distress costs on the pricing of reinsurance contracts.

*First*, using this model, we investigate the imperfections and limitations of the reinsurance market. We find that the pricing markup (or loading) and credit risk of a typical reinsurance contract will typically increase with the attachment point, even if the amount of coverage is small. We conclude that reinsurance should be used to cover the lower loss layers, and that CAT bonds should be issued to cover the larger, less probable loss layers. We then demonstrate that this strategy can in fact provide a desired coverage with cost and exposure at default significantly lower than those in other hedging strategies.

*Second*, we study how an optimal mix of reinsurance and CAT bond enables an insurer to deliver higher shareholder value in the presence of frictional costs stemming from financial distress and policyholders’ sensitivity to the insurer’s default risk. We find that an optimal mix of CAT bond and reinsurance allows an insurer to significantly increase shareholder value. We show that this is because the cost of hedging is lower under this strategy, which allows the insurer to purchase more protection and increase its capacity to profitably sell more insurance coverage. We also show that our results are consistent with Hagendorff et al.’s [2014] empirical observation that insurance firms that issue CAT bonds tend to have less exposure to catastrophe risks.

The article continues as follows. The next section presents the model framework, the formulation of optimization problems, and the numerical methods. The following section presents and discusses the numerical optimization results, and the final section summarizes our findings and concludes.

## MODEL FRAMEWORK

Let us consider a catastrophe insurance firm that seeks to maximize shareholder value. This firm can sell catastrophe coverage and can also consider a traditional reinsurance contract and/or a CAT bond. The insurer’s demand for hedging is driven by the desire to avoid financial distress costs and by the sensitivity of the insurance premium loading with respect to insolvency risk. Buying coverage is costly, but it allows the insurer to increase its risk-bearing capacity. In our model, the advantage of using reinsurance comes from the presumed existence of a beneficial business relationship between the insurer and the reinsurer, which reduces the cost of moral hazard. However, the reinsurer is vulnerable to default risk. Moreover, the premium loading charged by the reinsurer increases if the new contract exacerbates its expected cost of financial distress. On the contrary, the CAT bond is assumed to have no credit risk, but its investors require a substantial premium to compensate them for their exposure to moral hazard risk, which is present since CAT bond investors have limited capacity to monitor the CAT bond sponsor.

### Modeling the Insurer Equity and Assets

We consider a single-period model in which the insurer receives insurance premiums at time *t* = 0 and has to indemnify a stochastic underwriting loss *L**T* at time *t* = *T*. As in Krvavych and Sherris [2006], we model financial distress costs by means of the insurer’s deadweight losses incurred when the terminal net asset value is below the financial distress barrier. This threshold is assumed to be equal to an exogenously predetermined proportion *k**L* of the insurer’s initial asset value. We also assume limited liability, so that the terminal payoff to the shareholders is bounded below by zero. When the terminal asset value *A*_{T} is less than the value of insurance liabilities *L*_{T}, if the net asset value *A*_{T} − *L*_{T} is above the financial distress barrier *K*, then the firm is considered solvent and shareholders receive a liquidating dividend equal to *A*_{T} − *L*_{T}. However, if financial distress happens (i.e., *A**T* − *L**T* ≤ *K*), then the insurer suffers deadweight losses (1 − ω_{L}) as a proportion of the asset value. In this case, the terminal value of equity is given by max{ω;_{L}*A*_{T} − *L*_{T}, 0}. Therefore, the equity value at time *T* is given by

where *A*_{0} is the initial value of assets, and ω_{L} ∈ [0,1] and *k _{L}* ∈ [0,1] are exogenous constants.

The insurer’s assets include the initial capital *X*, plus the premium π_{P} received at *t* = 0 from the policyholders. The insurer can also employ two hedging instruments: a reinsurance contract that costs π_{RE} at *t* = 0 and imburses a cash amount R_{ET} at *t* = *T*, and a CAT bond that costs π_{CB} at *t* = 0 and pays a cash flow C_{BT} at *t* = *T*. Since insurance firms hold a large proportion of fixed-income securities in their portfolios, for simplicity, we further assume that the only other investment available to the insurer earns the risk-free rate *r*.^{4} Since hedging instruments are considered as assets for the purpose of financial accounting, the terminal asset value is

Our insurer’s demand for hedging is in part motivated by the desire to avoid financial distress, similar in spirit to other rationales put forward by Doherty [1991] and Froot and Stein [1998]. In Doherty [1991], risk aversion on the firm’s part is due to the presence of costs that are related to the variance of the firm’s primary cash flows, while in Froot and Stein [1998] a risk management incentive is derived from the convex cost of raising external capital for post-loss financing. In our model, financial distress costs subsume these kinds of losses. Moreover, as described in the next section, the sensitivity of the insurance premium loading with respect to insolvency risk, stemmed from policyholders’ risk aversion, constitutes another incentive for risk management.

### Modeling the Pricing of Insurance Policies

The insurance premium π^{P}, which is the amount paid at time *t* = 0 by the policyholders in exchange for catastrophic insurance, is of the form

where ℚ is the risk-neutral measure,^{5} *P*_{T} is the amount paid at time *T* to the policyholders,^{6} and δ_{P} is the loading the policyholders are willing to pay for this insurance contract. In other words, the market allows for “add ups” on the risk-neutral price.

As in Gatzert and Kellner [2014], we assume that the insurance company is not in a monopolistic position and cannot fix δ_{P} arbitrarily. On the contrary, we assume that the loading is endogenous and reflects the policyholders’ risk aversion. Specifically, we assume that δ_{P} decreases with an increase in the insurer’s objective default probability Pd. This is supported by the experimental results presented by Zimmer et al. [2009] and Zimmer et al. [2012], who find that the policyholders’ willingness to pay higher loadings decreases with the insurer’s default probability. Following Gatzert and Kellner [2014], we assume that the loading is given by^{7}

where ℙ is the real-world measure, is an exogenous constant denoting the maximum loading the policyholders are willing to pay if the insurer is default-free, and *q* represents the policyholders’ aversion to insolvency risk, which is assumed to be constant over time.

### Modeling the Pricing of the CAT Bond

The price paid at time *t* = 0 by the insurer for issuing the CAT bond is of the form^{8}

where δ_{CB} is the loading demanded by investors to compensate them for the inherent moral hazard risk stemming from the indemnity trigger feature.

Following Gatzert and Kellner [2014], we assume that the loading δ_{CB} is proportional to the ratio of the expected payment from the CAT bond (i.e., *E*^{ℙ}[CB_{T}]) to the expected loss of the insurer (i.e., *E*^{ℙ}[*L*_{T}]).^{9} The loading is thus of the form

where θ_{CB} is a constant capturing the investors’ exposure to moral hazard and is the minimum loading that investors would require in the absence of moral hazard risk. In Gatzert and Kellner [2014], the convexity of the cost function can be controlled by an additional parameter denoted by *k*. This parameter is set to 1 in our work, hence, we assume a linear cost function. We assume the CAT bond is not subject to both counterparty risk and basis risk.

### Modeling the Pricing of the Reinsurance Contract

In contrast to CAT bonds, credit risk is inherent in reinsurance contracts. Let denote the reinsurer’s asset value prior to selling a reinsurance contract to the insurer, and let π_{RE} denote the price paid at time *t* = 0 by the insurer for that reinsurance contract. The initial value of the reinsurer’s total assets is given by + π_{RE} and is assumed invested at the risk-free rate. The terminal payoff to the insurer is given by

where is the payoff that the insurer receives from a full faith and credit reinsurer, and ℒ_{T} denotes the terminal value of the reinsurer’s liabilities that are senior to the new contract.

We assume that the reinsurer takes financial distress costs into account in his pricing of the reinsurance contract, as follows:

8where is the competitive loading, θ_{RE} represents the reinsurer’s exposure to moral hazard, R_{E0} is the actuarially fair value of the contract, and is the price of the reinsurance contract that would leave the reinsurer’s equity unchanged. In other words, we have

where denotes the reinsurer’s equity prior to entering into the new contract and denotes the reinsurer’s equity after the sale of the new contract at price π. Note that the reinsurer’s equity is modeled in the same fashion as in Equation (1), with terminal assets given by ( + π_{RE})*e*^{rT}, terminal liabilities given by , and financial distress parameters denoted by ω_{ℒ} and .

Equation (8) means that the reinsurance firm would like to offer the loading , which includes a moral hazard premium but not financial distress costs. However, the reinsurance firm will charge a higher loading if this price decreases shareholder wealth once distress costs are taken into account. In other words, the reinsurer’s reservation price is such that shareholder value does not decrease.^{10} As shown later, as financial distress costs are prevalent, the reinsurer demands higher loadings for selling protection against high layers of losses that have low occurrence probabilities.

### Comparing CAT Bonds with Traditional Reinsurance

Doherty and Smetters [2005] find that the cost of moral hazard is significantly lower when the insurer and the reinsurer are affiliates. Therefore, we assume moral hazard risk to be significantly smaller for the reinsurer than for the CAT bond investors (or θ_{RE} < θ_{CB}), the reasons being that the reinsurer’s ability to monitor the insurer is much greater than capital market investors’ ability to carry out surveillance on the CAT bond sponsor, and that a long-term business relationship is assumed to exist between the insurer and the reinsurer.

To sum up, the reinsurance pricing is mainly driven by the risk aversion embedded in financial distress costs, whereas the CAT bond pricing is impacted by moral hazard. Furthermore, we assume no default risk for the CAT bond but credit vulnerability for reinsurance. This feature depicts the differences between these two hedging instruments in our model.

We reckon that other factors might affect the pricing. For instance, Zhu [2011] demonstrates that ambiguity aversion could explain why CAT bond investors require higher loadings when the probability of the first loss is very small. Also, Perrakis and Boloorforoosh [2013] show that CAT bond investors should demand higher loadings if the insurer’s loss is correlated with traditional financial instruments. In our model, we implicitly assume that this correlation is negligible.^{11}

Note also in our model, the pricing markups are endogenous in the sense that these are obtained from well-defined pricing mechanisms. We define an exogenous functional form that links the insurer’s default probability markups to a maximum exogenous loading, and an exogenous parameter that represents the policyholders’ aversion to the risk of insolvency.

### Modeling the Insurer Liabilities

Following Lo et al. [2013] and numerous others, we model the market value of insurance liabilities using Merton’s [1976] jump-diffusion model. This approach explicitly models catastrophic events by adding jumps in liabilities that arrive randomly according to a Poisson process. In this setting, the real-world dynamics of the insurer’s liabilities is given by

10where μ_{L} and σ_{L} respectively denote the drift and the volatility of the geometric Brownian motion component, denotes a ℙ-Wiener process and *J _{t}* is the following piecewise constant process

In the above, *N _{t}* is a ℙ homogeneous Poisson process with intensity parameter λ ≥ 0, and

*Y*− 1 is the size of the

_{j}*j*-th jump. In other words, if the

*j*-th jump occurs at time

*t*, the liabilities jump from

*L*to

_{t−}*L*=

_{t}*Y*. Using the fact that

_{j}L_{t−}*J*

*t*− λ

*mt*with

*m*=

*E*

^{ℚ}[

*Y*] − 1 is a ℚ-martingale, Merton [1976] shows that the solution of the stochastic differential equation (10) under the risk-neutral measure ℚ is given by

where denotes a ℚ-Wiener process and *r* is the risk-free rate. Assuming that jumps are diversifiable, following Merton [1976], the {*Y _{j}*} are i.i.d. according to a lognormal distribution

*LN*(

*a*,

*b*

^{2}).

The market value of the reinsurer’s liabilities is also modeled using the jump-diffusion model described above, and we allow for the possibility that the insurer’s and reinsurer’s liabilities may be correlated, as follows

13In the subsequent sections, we present the hedging strategies, then set up the optimization problems, and finally present and discuss the results obtained by using numerical methods.

### Analyzing the Hedging Strategies

We focus on limited (or truncated) excess-of-loss (XL) hedging strategies. The strategies are all identical in terms of payoff function and differ only in terms of cost and credit risk. The four key strategies stem from the following reinsurance contract and CAT bond^{12}

where *H*_{RE} and *H*_{CB} denote the attachment point for the reinsurance contract and the CAT bond respectively, and *M*_{RE} and *M*_{CB} denote the layer limit for the reinsurance contract and the CAT bond respectively.

The four strategies are displayed in Exhibit 1. Strategy R_{E} uses reinsurance but no CAT bond, whereas Strategy CB uses a CAT bond but no reinsurance. Strategy RECB uses a reinsurance contract with attachment *H*_{RE} and limit *M*_{RE}, and a CAT bond with attachment *H*_{RE} + *M*_{RE} and limit *M*_{CB}. The resulting payoff of this strategy is the following XL function

Under Strategy RECB, therefore, lower losses are covered with reinsurance and higher losses are hedged with a CAT bond. Although there is no reason to offer CAT bonds to cover low losses, Strategy CBRE does exactly the opposite and its payoff function is

16Note that Equations (15) and (16) describe the payoff in the case of a default-free reinsurer. The actual payoff (with default risk) is RE_{T} + CB_{T}, with RE_{T} given by Equation (7).

We see that all strategies follow a payoff function identical to that of a single XL contract (see Exhibit 2 for an example). Since they all span the same set of payoff functions, it is legitimate to compare these four hedging strategies. This is an important point, since comparing hedging strategies that use dissimilar “building-blocks” for the payoffs does not pin down and isolate the beneficial effect of the hedge’s cheaper price and reduced credit risk.

### Minimization of the Insurer Hedging Cost

How can the insurer optimally combine reinsurance with a CAT bond to minimize its hedging cost? Seeking answers to this question is of paramount importance since it will ultimately help us to understand how the insurer can increase shareholder value by optimally combining a reinsurance contract with a CAT bond.

Let us suppose that the insurer seeks to obtain an XL hedging contract with attachment *H* and layer limit *M*.^{13} Under Strategy RE (reinsurance) or Strategy CB (CAT bond), the insurer is constrained to use a single hedging instrument. Intuitively, both of these strategies will likely lead to the insurer having to pay a significant loading for the prices. For instance, using reinsurance alone might apply too much pressure on the reinsurer, which could result in a contract with a significant probability of default, and a higher price to compensate the reinsurer for bearing financial distress costs. Otherwise, using only a CAT bond might result in the insurer having to pay a high price loading to compensate investors for its exposure to moral hazard.

To overcome this problem, the insurer could instead use Strategy CBRE, which hedges low losses through the issuing of a CAT bond and covers higher losses with reinsurance. The objective of the insurer is then to use this strategy to replicate the desired XL contract (i.e., with attachment *H* and limit *M*) at the lowest price possible. Given that the CAT bond is assumed default-free, we suppose that the insurer imposes a constraint on the hedge’s probability of default HPD, which is given by the probability that the reinsurance instrument defaults

For our exercises, we suppose that the insurer requires the hedge’s default probability Hpd to be below a predetermined level *p* (e.g., 0.2%). Under Strategy CBRE, the insurer then has to determine the optimal amount of reinsurance *M*_{RE}. To achieve this, the insurer solves^{14}

Alternatively, the insurer could use Strategy RECB, which covers small losses with reinsurance and covers high losses through the issuing of a CAT bond. The corresponding problem is then

19For *p* sufficiently small, it is interesting to note that minimizing the hedging cost is equivalent to minimizing the *effective loading* δ_{EFF} paid by the insurer. Interestingly, δ_{EFF} can be written as^{15}

where π_{RE} + π_{CB} is the total cost of the hedge, RE_{0} = *E*^{ℚ}[RE_{T}]*e*^{−}^{rT} is the actuarially fair value of the reinsurance component, CB_{0} = *E*^{ℚ}[CB_{T}]*e*^{−rT} is the actuarially fair value of the CAT bond component, δ_{RE} is the loading paid to the reinsurer, and δ_{CB} is the loading paid to the CAT bond’s investors. Equation (20) shows that the effective loading is a weighted average of the loading paid to the reinsurer and that paid to the CAT bond’s investors. The goal of the insurer is to minimize this weighted average, and we will see later that Strategy RECB is the optimal one.

### Maximization of the Insurer Shareholder Value

By how much can the insurer increase its shareholder value by mixing reinsurance and CAT bond? This problem is obviously related to the minimization of the hedging cost. In fact, the linkage between these problems is evident if we decompose the insurer’s equity value into

21where *X* is an exogenous constant denoting the initial equity capital available to the insurer,^{16} *P*_{0} = *E*^{ℚ}[*P*_{T}]*e*^{−}^{rT} is the actuarially fair value of the insurance policies, and CDF denotes the expected cost of financial distress. By definition, this quantity is the difference between the equity value *S*_{0} priced as if financial distress were devoid and the actual (true) equity value when the prevalent financial distress is taken into account.

The insurer goal is to simultaneously determine the optimal risk exposure *L*_{0} and the optimal hedging strategy. We assume that the insurer is free to choose the initial value *L*_{0} of the loss process. In practice, it might be true that the insurer has little control over the demand for insurance, which means that it could be more realistic to impose bounds on the value of *L*_{0}. Nevertheless, we let *L*_{0} be an unconstrained parameter, since our goal is to study how the underwriting *capacity* of the insurer is enhanced by an optimal mix of reinsurance and CAT bond. Note that the insurer is assumed to have no pre-existing risk exposure, in that the total value of insurance liabilities at time 0 is *L*_{0}. We assume the volatility of loss σ_{L} to be an exogenous constant that is independent of the risk exposure *L*_{0}. We thus neglect potential diversification effects and consider the insurer underwriting portfolio to contain only one representative type of catastrophic risk.

Equation (21) allows us to see how the values of the controls (or decision variables) affect the equity value *S*_{0}. First, an increase in the risk exposure *L*_{0} increases equity by increasing *P*_{0}. However, an excessively high value of *L*_{0} also decreases equity by increasing the distress costs CDF and decreasing the loading δ_{P} that policyholders are willing to pay. The underwriting capacity of the insurer is thus limited. However, the insurer can use hedging instruments, which can increase equity by increasing both δ_{P} and *P*_{0} and also by decreasing Cdf. Second, hedging also decreases equity by − δ_{CB} CB_{0} − δRE RE_{0}, justifying the insurer motive to find an optimal mix of reinsurance and CAT bond that minimizes the effective loading δ_{EFF} for the overall strategy.

The formulation of the optimization problem depends on the hedging strategy used by the insurer. For Strategy RE (reinsurance only), the optimization problem takes the form

22For Strategy CB (CAT bond only), the optimization problem takes the form

23For Strategy RECB (optimal mix of reinsurance and CAT bond), the optimization problem is

24Since the results from the problem of minimizing hedging costs show that Strategy CBRE is not optimal relative to Strategy RECB, we do not consider Strategy CBRE when studying the maximization of shareholder value.

Note that no solvency constraints are imposed on the insurer. Due to the fact that financial distress costs and policyholders’ risk aversion make it optimal for the insurer to have a near-zero default probability, such exogenous constraints are not necessary for our model. This feature aligns with experimental results by Zimmer et al. [2012], who find that policyholders’ risk aversion induces insurers to bear no default risk. We find from our experiments that imposing different types of solvency constraints (as in Bernard and Tian [2009], for instance) does not change our results.

### Numerical Methods

Since they do not have known closed-form solutions, we resort to numerical methods to solve the optimization problems presented above. Note for instance, that Equation (3) can be written as

25The insurance premium π_{P} appears on both sides of the equation and cannot be separated, implying that a fixed-point algorithm must be used to solve for the value π_{P}.

To compute the optimization problem objective functions, we use the Monte Carlo approach, with quadratic resampling and antithetic variates to improve statistical accuracy (see, e.g., Huynh et al. [2008]). The optimization problems are solved using Matlab’s pattern search algorithm. This is a direct search method, introduced by Hooke and Jeeves [1961], which can search through more than one basin of attraction. Moreover, this method does not require the gradient of the problem to be optimized and is suitable even if the objective function has discontinuities.

To reduce the odds of finding a local minimum, we use a multistart approach. For instance, for the optimization problem given by Equation (24), we start by generating a relatively large (e.g., 300,000) random sample for the value of the input vector (*L*_{0}, *H*_{RE}, *M*_{RE}, *M*_{CB}). By virtue of the constraint *H*_{CB} = *H*_{RE} + *M*_{RE}, the parameter *H*_{CB} is not in the input vector. The objective function is then calculated for each guess, and a small number (say 20) of the best guesses are each used as the starting point of the pattern search algorithm. The best found minimum is then taken as the solution.

For our optimization purposes, the parameters *H*_{RE}, *M*_{RE}, *M*_{CB} are scaled by *L*_{0}, since this parameter determines their order of magnitude. In other words, each parameter is expressed as a multiple *xL*_{0} and the optimization bounds are on *x*. This approach also greatly facilitates the definition of the search space for the optimization algorithm.

## OPTIMIZATION RESULTS

In this section, we study how an insurer with exposure to catastrophe risk can optimally combine a traditional reinsurance contract with a CAT bond to minimize the hedging cost and ultimately maximize shareholder value. As indicated earlier, we compare four hedging strategies (see Exhibit 1) with XL payoffs that differ only in terms of pricing and credit risk. One strategy uses reinsurance alone (RE); one relies only on a CAT bond (CB); one chooses a CAT bond to cover smaller losses in conjunction with a reinsurance contract to cover larger losses (CBRE); and one employs a reinsurance contract to cover smaller losses and a CAT bond to protect against larger losses (RECB). We determine which strategy allows the insurer to obtain the desired coverage at the lowest cost, and whether this leads to significant shareholder value creation for the insurer. We also perform sensitivity analyses to identify the main factors contributing to the effectiveness of an optimal mix of reinsurance and CAT bond.

### Parameters

The baseline input parameters are presented in Exhibit 3, where the attachment *H* and the layer limit *M* of the desired XL contract values are those obtained from the shareholder value-maximizing exercise. Since it is extremely difficult to find real data to estimate the parameters involved in our model, we borrow these from other authors. The values for the financial distress barriers are drawn from Krvavych and Sherris [2006]. The deadweight cost of financial distress is taken from Andrade and Kaplan [1998], who document that this cost can be 20% of the market value of assets for financial companies. Due to policyholders’ aversion to insolvency risk, this cost is likely to be higher for insurance companies. According to Cummins and Weiss [2009], CAT bonds are nowadays competitive with respect to traditional reinsurance; therefore, we set the minimum CAT bond loading equal to the minimum reinsurance loading. Most other baseline parameters are either drawn from Gatzert and Kellner [2014] or chosen for the purposes of our comparative statics experiments. We also conduct robustness tests on all input parameters to validate the generality of our findings.

### Investigating the Limitations of the Reinsurance Market

In the next sections, we present results showing that the optimal hedging strategy is to use reinsurance to cover lower layers of losses and issue a CAT bond to cover higher layers of losses. The present section dissects the main mechanism driving these results.

The results presented in Exhibit 4 illustrate the imperfections and limitations of the reinsurance market, by showing how the loading and credit quality of the reinsurance contract vary with the trigger probability ℙ(*L*_{T} ≥ *H*_{RE}). The main conclusion is that reinsurance is more expensive and less creditworthy when it is used to cover low probability, high severity layers of losses.

First, from Panels (a) and (c) of Exhibit 4 it can be seen that the loading generally increases as the trigger probability decreases,^{17} especially when the insurer’s and reinsurer’s losses are strongly correlated, consistent with empirical evidence in Froot [2001]. The reason is that in our model, the reinsurer takes financial distress costs into account in the determination of his reservation price.^{18} Because the portfolios of liabilities of the insurer and reinsurer are correlated, it is more costly for the reinsurer to provide protection against low probability, high severity losses. Moreover, we can see from Panels (b) and (d) of Exhibit 4 that the credit quality of the reinsurance contract decreases with the trigger probability. A reinsurance contract covering low probability, high severity losses is more vulnerable to credit risk (more likely to default). This is especially true when the correlation ρ between the insurer’s and the reinsurer’s losses is high, and when the layer limit *M*_{RE} is large (see Panels (d)).

Interestingly, the loading and the credit risk that go with the reinsurance contract can be substantial even when the layer limit is very small (see Panels (a) and (b) of Exhibit 4). Because of this feature, even if a small layer limit were transferred to several different reinsurers, each contract loading and credit risk would still be high as long as the trigger probability is small. Our pricing framework does not necessarily encourage a mix of several reinsurance contracts with different reinsurers. Nevertheless, it does promote a mix of a reinsurance contract with a CAT bond. As a matter of fact, the loading of the CAT bond (see Equation (6)) is a (convex) function of the *amount* of coverage purchased and does not depend directly on the attachment point (or trigger probability). Moreover, if we were to incorporate risk aversion in the CAT bond pricing framework, the CAT bond loading would then decrease with the trigger probability, thereby making the CAT bond to complement reinsurance to a greater extent.

In sum, the loading and credit risk associated with the reinsurance contract typically will increase with the attachment point, even if the amount (or layer limit) of the protection is very small, explaining why it makes sense to use reinsurance for the lower layers of losses and issue a CAT bond for the larger, less probable loss layers. In the next sections, we show that this strategy is optimal using two approaches: first, by demonstrating that it provides the desired coverage for a lower cost and exposure at default, and second, by demonstrating that it expands the insurer underwriting capacity and yields its shareholders higher values.

### Results for the Minimization of the Insurer Hedging Costs

In this section, we investigate how a reinsurance contract and a CAT bond can be combined by an insurer so as to minimize the price of a desired limited excess-of-loss (XL) coverage.

Intuitively, the optimal proportion of reinsurance for the insurer to purchase depends on the correlation between the insurer’s and the reinsurer’s losses. Exhibit 5 exhibits the results for the four hedging strategies (see Exhibit 1) for different values of the correlation ρ. We report the results for the reinsurance layer limit *M*_{RE}, which is related to the CAT bond layer limit *M*_{CB} by the relation *M*_{RE} + *M*_{CB} = *M*, where *M* is the layer limit of the desired XL contract to be synthesized. By definition, we always have *M*_{RE} = *M* and *M*_{CB} = 0 for Strategy RE (reinsurance), whereas we have *M*_{RE} = 0 and *M*_{CB} = *M* for Strategy CB (CAT bond). For Strategies CBRE and RECB, the optimal values of *M*_{RE} are respectively derived by solving the optimization problems described by Equations (18) and (19).

Exhibit 5 also presents the corresponding proportion of reinsurance where RE_{0} is is the reinsurance contract actuarially fair value and CB_{0} is the CAT bond actuarially fair price. The hedge total cost Π, the hedge’s effective loading δ_{EFF} (see Equation (20)), and the hedge’s objective probability of default Hpd (see Equation (17)) are also presented in Exhibit 5.

First of all, Exhibit 5 shows that Strategy RECB results in lower hedge prices Π, and yields effective loadings δ_{EFF} that are about twice as small as those for other strategies. These results indicate that hedging small losses with reinsurance and higher losses by issuing a CAT bond (Strategy RECB) is optimal, in the sense that it provides the desired coverage for a lower cost. Due to the constraint HPD ≤ 0.2% on the hedge’s probability of default, Strategy CBRE actually leads to higher prices than Strategy RE. For Strategy RE, the default probability is in fact always above 0.2%, which is the upper bound imposed in Strategies RECB and CBRE.

The results in Exhibit 5 also show that the optimal proportion of reinsurance decreases with an increasing value of the correlation ρ between the insurer’s and the reinsurer’s losses. Under Strategy RE (reinsurance), the hedge’s default probability, its loading, and its cost all increase with an increasing value of ρ. This is because a higher correlation ρ exacerbates the reinsurer’s risk of insolvency and expected cost of financial distress. We see from Equation (8) that the reinsurer will require a larger loading in such circumstances. When CAT bonds are used in tandem with reinsurance, we conclude from these results that correlation not only limit the efficiency of reinsurance but also reduce the demand for it.

Exhibit 6 illustrates the results for all of the studied hedging strategies under different values of the reinsurer’s risk exposure ℒ_{0}. Again, we notice that using reinsurance for lower layers of losses and the CAT bond for higher layers of losses (Strategy RECB) gives rise to both lower hedging costs Π and effective loadings δ_{EFF}. The optimal proportion of reinsurance decreases with an increasing value of the reinsurer’s risk exposure ℒ_{0}. When we use reinsurance only (Strategy RE), the hedge’s default probability, its loading, and its cost all increase with an increasing value of ℒ_{0}. This is explained by the fact that a reinsurer with greater current risk exposure has less capacity to take on more risk. The reinsurer will therefore charge a higher loading to compensate the increased expected cost of financial distress. Naturally, a reinsurer with high pre-existing risk exposure ℒ_{0} being less creditworthy is pricier in loadings. We conclude that the limited capacity of reinsurers may curtail the usefulness of this market, and may also reduce the demand for reinsurance when CAT bonds are readily available.

We next investigate how the insurer’s risk exposure *L*_{0} affects the demand for reinsurance when CAT bonds are available. The lefthand graph of Exhibit 7 illustrates how the optimal proportion of reinsurance under Strategy RECB decreases with an increasing risk size *L*_{0}, for three different correlations ρ. When the exposure *L*_{0} is relatively small and the correlation ρ is low, the reinsurance component constitutes around 80% of the hedge. However, for larger risks and higher correlations, the optimal proportion of reinsurance drops to around 45%. Therefore, reinsurance is an important part of the hedge when the insurer’s risks are relatively small and weakly correlated to the reinsurer’s risks.

We also study how the attachment point *H* of the desired XL coverage impacts the optimal proportion of reinsurance under Strategy RECB. The righthand graph of Exhibit 7 illustrates how the optimal proportion of reinsurance decreases with an increasing attachment point *H*, for three values of correlation. When the attachment point is relatively low and the correlation ρ is weak, we see that the reinsurance component constitutes about 75% of the hedging strategy. However, the optimal proportion of reinsurance quickly decreases with an increasing value of the attachment point, which indicates that it is optimal to use CAT bonds when the desired attachment point is high. This also explains why hedging smaller losses with reinsurance and higher losses with a CAT bond (Strategy RECB) is optimal.

Summing up, our results demonstrate that a combination of a CAT bond with a reinsurance contract can be optimal in the sense that it provides the desired coverage for a significantly lower cost, in accord with an empirical observation by Härdle and Cabrera [2010]. We show that the optimal strategy is to use the CAT bond to hedge higher layers of losses and reinsurance to cover lower layers of losses. This strategy is optimal mainly because in the presence of financial distress costs, the reinsurer charges higher loadings for protection against higher layers of losses. Our findings complement the results of Nell and Richter [2004], who find substitution effects between reinsurance and a parametric CAT bond for large losses only. We confirm the intuition discussed in Cummins and Trainar [2009], that traditional reinsurance is very useful when risks are relatively small and weakly correlated, but that the efficiency of this market breaks down as the magnitude of potential losses and the correlation of risks increase.

### Results for the Maximization of the Insurer Shareholder Value

We next study how a combination of a reinsurance contract with a CAT bond translates into higher shareholder values for the insurer.

Exhibit 8 exhibits shareholder maximization results for different values of the insurer’s loss volatility σ_{L}, for each of the three hedging strategies under consideration (CB, RE, and RECB). For all strategies, the optimal values of the risk exposure *L*_{0}, the reinsurance attachment *H*_{RE}, the reinsurance layer limit *M*_{RE}, the CAT bond attachment *H*_{CB}, and the CAT bond layer limit *M*_{CB} are presented. For Strategy RE (reinsurance only), we obtain these parameters by solving the optimization problem in Equation (22). For Strategy CB (CAT bond only) and Strategy RECB (reinsurance and CAT bond), these parameters are derived by solving the problems in Equations (23) and (24), respectively. Exhibit 8 also presents the maximum shareholder value *S*_{0} = *E*^{ℚ}[*S*_{T}]*e*^{−rT} (see Equation (1)), the insurer’s default probability Pd (see Equation (4)), and the other outputs that are required to decompose the shareholder value into (see Equation (21))

where *X* = $100*M* denotes the insurer’s initial equity capital (see Exhibit 3), δ_{P} is the insurance loading (see Equation (4)), δ_{EFF} is the hedge effective loading (see Equation (20)), *P*_{0} = *E*^{ℚ}[*P*_{T}]*e*^{−}^{rT} is the actuarially fair value of insurance policies (see Equation (3)), Cdf is the insurer’s expected cost of financial distress, and the sum RE_{0} + CB_{0} is the actuarially fair value of the hedge, with RE_{0} = *E*^{ℚ}[RE_{T}]*e*^{−}^{rT} and CB_{0} = *E*^{ℚ}[CB_{T}]*e*^{−}^{rT} (see Equation (14)). Recall that the expected cost of financial distress CDF is defined as the reduction in the equity value *S*_{0} if a state of financial distress occurs. CDF is the difference between the equity value in the absence of financial distress (i.e., with ω_{L} = 1) and the equity value when financial distress kicks in.

Results from Exhibit 8 show that an optimal mix of reinsurance and CAT bond (Strategy RECB) enables the insurer to gain shareholder values *S*_{0} that are about 2%–3% higher than obtained using the other strategies. This is mainly because the optimal risk exposure *L*_{0} is significantly larger under this strategy, which leads to a higher fair value of insurance *P*_{0}. Moreover, the policyholders’ loading δ_{P} is always close to its upper bound (10%), which is due to the fact that the insurer’s probability of default Pd is, most of the time, very close to zero. In this setting, it is optimal for the insurer to be nearly default free, which explains why the actuarially fair value of insurance policies *P*_{0} is always so close to the value of the insurer’s risk exposure *L*_{0}.

In addition, we note that expected financial distress costs CDF are invariably very small (~$0.05*M*). The insurer also uses more hedging under Strategy RECB, buttressed by the hedge’s fair value RE_{0} + CB_{0}. As shown in Exhibit 8, the ratio of the hedge’s fair value RE_{0} + CB_{0} to the optimal risk exposure *L*_{0} is always higher under Strategy RECB, which means that the insurer covers a larger fraction of its losses when this strategy is used.^{19} Despite this, we see that the hedge’s effective loading δ_{EFF} is always smaller for this strategy. Exhibit 8 shows that an optimal mix of reinsurance and CAT bond enables the insurer to obtain a higher equity value due to the fact that the cost of hedging is lower, which allows the insurer to purchase more protection, thereby increasing its underwriting capacity (i.e., the optimal value of *L*_{0}).

These findings are exhibited in Exhibit 9, where the left-hand graph plots the maximum shareholder value *S*_{0} against the insurer’s loss volatility σ_{L}, and the right-hand graph plots the optimal insurer’s risk exposure *L*_{0} against σ_{L}. In all strategies, both shareholder value and risk exposure decrease with increases in loss volatility, because insurance policies are then riskier and more expensive to cover. More importantly, it can be seen in Exhibit 9 that Strategy RECB dominates the other strategies in terms of shareholder value *S*_{0} and insurance capacity *L*_{0}, which further confirms the results displayed in Exhibit 8. We conclude that both low-volatility and high-volatility insurers would benefit from combining reinsurance and CAT bonds.

Exhibit 10 displays the results from the maximization of shareholder value for all of our hedging strategies, for different values of the reinsurer’s risk exposure ℒ_{0}. These results further confirm that an optimal mix of reinsurance and CAT bond yields higher shareholder value, once again, because the lower cost of hedging under this strategy enables the insurer to increase its optimal risk exposure *L*_{0} while staying out of financial distress. Under Strategy RECB, we clearly see that the optimal reinsurance layer limit *M*_{RE} decreases with an increasing value of the reinsurer’s risk exposure ℒ_{0}, meaning that less reinsurance is used when the reinsurer has high pre-existing risk exposure.

Recall that the same finding is obtained in the previous section, in which we tackle the problem of minimizing the hedging cost. Again, the reason is that a reinsurer with a greater pre-existing risk exposure has less capacity to take on additional catastrophe risk without increasing its expected cost of financial distress, and will therefore charge a larger premium. Also, a reinsurer with a high risk exposure ℒ_{0} is less creditworthy. It can be seen from Exhibit 10 that, if the reinsurer’s risk exposure is *L*_{0} = $500*M*, then the optimal reinsurance layer limit is *M*_{RE} = 0 under Strategy RECB, meaning that no reinsurance is purchased by the insurer. In this particular case, the insurer’s optimal risk exposure *L*_{0} and optimal hedging parameters *H*_{RE}, *M*_{RE}, *H*_{CB}, *M*_{CB} are then identical to those obtained under Strategy CB (CAT bond only), which means that Strategy RECB converges to Strategy CB as the value of ℒ_{0} increases.

These findings are evident from Exhibit 11, where the left-hand graph plots maximum shareholder value *S*_{0} against reinsurer’s risk exposure ℒ_{0}, and the right-hand graph shows the optimal insurer’s risk exposure *L*_{0} against ℒ_{0}. If the reinsurer’s risk exposure is sufficiently small (ℒ_{0} ~ $200*M*), then Strategy RE (reinsurance) and Strategy RECB converge to the same values of *S*_{0} and *L*_{0}, because it is then optimal to rely entirely on reinsurance. However, as ℒ_{0} increases, the maximum shareholder value and optimal risk exposure remain higher under Strategy RECB, and eventually converge to the solution obtained under Strategy CB. It is optimal to use 100% reinsurance if the reinsurer’s risk exposure ℒ_{0} is sufficiently small, whereas it is optimal to resort to 100% CAT bonds if ℒ_{0} is sufficiently large. This shows that CAT bonds complement reinsurance very well, especially when the reinsurer has limited capacity.

Finally, to check whether our model can explain an empirical observation documented by Hagendorff et al. [2014], that CAT bonds tend to be issued by insurance firms with low risk exposure, we study how the insurer’s risk exposure *L*_{0} is related to the optimal proportion of CAT bond required. We again derive the optimal hedge parameters *H*_{RE}, *M*_{RE}, *H*_{CB}, *M*_{CB} by solving the optimization problem in Equation (24), but in this case with an additional constraint that . The insurer’s risk exposure is now considered exogenous and cannot be changed by the insurer. Exhibit 12 illustrates the optimal proportion of CAT bond and the hedge’s relative fair value against the insurance demand available to the insurer. There is less need to hedge when is small, which results in very small hedges relative fair values (~5%). We also note that, as observed by Hagendorff et al. [2014], a larger proportion of CAT bond is then utilized for the hedge, and this proportion decreases with an increasing . This is because the insurer only needs to hedge higher layers of losses when the value of is small. Such reinsurance contracts tend to be very expensive, making it optimal to issue a CAT bond.^{20}

To sum up, our results reveal that an optimal mix of CAT bond and reinsurance allows the insurer to deliver a higher shareholder value because hedging costs are shown to be cheaper. The results in this section show that this strategy allows the insurer to purchase more protection so as to increase its capacity to sell a larger amount of risk exposure in the insurance market. CAT bonds thus complement reinsurance very well, especially when the capacity in the reinsurance industry is limited. Both low-volatility and high-volatility insurers would benefit from mixing reinsurance and CAT bonds. Further, our results are in line with Hagendorff et al. [2014]’s empirical findings that CAT bonds tend to be issued by insurance firms with less catastrophe risk exposure and less need to hedge. The rationale we provide for this is that insurers with a low risk exposure need only to hedge higher layers of losses, which tend to be very expensive to cover by using traditional reinsurance.

## CONCLUSION

This article examines how an insurance firm covering catastrophe risk can optimally combine a traditional reinsurance contract with a CAT bond by minimizing its hedging cost and maximizing its shareholder value. To this end, we compare four hedging strategies, all of which have a limited excess-of-loss payoff function. The first strategy uses reinsurance only; the second uses a CAT bond only; the third employs a CAT bond to cover small losses and a reinsurance contract to cover larger losses; and the fourth strategy resorts to a reinsurance contract to cover small losses and a CAT bond to cover larger losses.

Our results indicate that the fourth strategy is optimal with respect to the minimization of the hedging cost and also with respect to the maximization of shareholder value. Our results demonstrate that mixing a CAT bond with a traditional reinsurance contract provides the desired coverage for a lower cost. We show that this strategy is optimal due to the fact that, with financial distress costs present, the reinsurer charges higher loadings to cover higher layers of losses, even if the amount of coverage is small. Moreover, credit risk is typically higher for such reinsurance contracts. It is therefore optimal to hedge smaller losses with reinsurance and higher losses through the issuing of a CAT bond. Our results also indicate that traditional reinsurance is very useful when risks are relatively small and weakly correlated. However, we find that the efficiency of the reinsurance market breaks down as the magnitude of potential losses and the correlation of risks increase, and that CAT bonds should be favored in such circumstances.

We also show that an optimal mix of reinsurance and CAT bond allows the insurer to deliver a higher shareholder value. This is because hedging is cheaper under this strategy, which allows the insurer to purchase more protection so as to increase its risk-bearing capacity. In addition, we show that the optimal proportion of reinsurance to purchase depends crucially on the pre-existing risk exposure of the reinsurer. For low-risk reinsurers it can be optimal to use only reinsurance, whereas for high-risk reinsurer it can be optimal to use only a CAT bond. Reinsurance and CAT bonds thus complement each other very well, especially when the reinsurance industry has a limited capacity.

Moreover, we find that a mix of reinsurance and CAT bond remains optimal for different values of the insurer’s loss volatility. In other words, both low-volatility and high-volatility insurers would benefit from combining reinsurance and CAT bonds. Finally, we also find that insurance firms with less exposure to catastrophe risk should issue a CAT bond rather than purchase reinsurance, assuming that the fixed cost of issuing a CAT bond is not discriminately prohibitive for such firms. The reason is because such firms need only to hedge higher layers of losses, which are very expensive to cover with reinsurance.

Unlike previous works that study how to combine reinsurance and alternative risk transfer (ART) instruments, we explicitly model the credit risk of the reinsurer and the effect of financial distress on the pricing of reinsurance contracts. We find that explicitly modeling these features is crucial in determining the optimal portion of reinsurance to purchase, enabling us to determine the usefulness of reinsurance and to include the effects of correlation, loss exceedance probability, and risk exposure on the pricing of reinsurance.

## ENDNOTES

We thank the Montréal Exchange, the Institute de Finance Mathématique de Montréal (IFM2), the Royal Bank of Canada, the Fonds Conrad Leblanc, the Autorité des marchés financiers, and the Social Sciences and Humanities Research Council of Canada for their financial support. We thank participants at the World Risk and Insurance Economics Congress (2015) in Munich, Germany, the International Finance and Banking Society (2015) Conference in Hangzhou, China, and the finance seminar of the Shanghai University of Finance and Economics (2015) in Shanghai, China, and the International Research Meeting in Business and Management (2017) in Nice, France. We appreciate the valuable comments and suggestions from Marie-Claude Beaulieu, Alexander Braun, Robert Dickson, Hélyoth Hessou, Bertrand Maillet, Duc Khuong Nguyen, Hato Schmeiser, Issouf Soumaré, Christoph Wegener, Min-Teh Yu, Xueying Zhang, and Wenge Zhu. We also thank Yoann Racine for excellent research assistance.

↵

^{1}See “Q2 2017 Catastrophe Bond & ILS Market Report” by ARTEMIS.↵

^{2}See Towers Watson [2010] for a discussion on CAT bonds credit risk in the context of a total return swap.↵

^{3}Krvavych and Sherris [2006] investigate the demand for change-loss reinsurance when the insurer’s objective is to maximize shareholder value under a solvency constraint and in the presence of frictional costs. Yow and Sherris [2008] show that, if the insured parties are risk-averse, there is a trade-off between improving the firm’s solvency condition and incurring costs of holding economic capital.↵

^{4}This is justified as it allows us to focus on the liability side. An interesting extension could be to allow the insurer to invest in risky assets. See Jang and Kim [2015] and references therein for non-CAT bond frameworks that combine reinsurance and financial asset allocation.↵

^{5}Following Yow and Sherris [2008] and Gatzert and Kellner [2014] among others, we assume the market to be arbitrage-free, such that the risk-neutral measure exists without necessarily being unique for our case.↵

^{6}We assume that policyholders are not affected by financial distress. Later, we show that this assumption does not matter much since it is optimal for the insurer to have a near-zero probability of financial distress.↵

^{7}Note that δ_{P}is bounded below by −1 to ensure that π_{P}≥ 0.↵

^{8}For examples of CAT bonds risk-neutral valuation, see Vaugirard [2003], Lai et al. [2014], Têtu et al. [2015].↵

^{9}Unlike Lee and Yu [2002], who model moral hazard by assuming that the insurer relaxes its settlement policy once the accumulated loss is close to the trigger, thus causing an increase in future expected losses, we follow Gatzert and Kellner [2014] to model moral hazard in a reduced form. As explained in the next section, the pricing of the reinsurance contract takes both financial distress and moral hazard into account. A full endogenization of moral hazard would make the pricing of the reinsurance contract prohibitively difficult from a numerical standpoint.↵

^{10}We could also assume that the reinsurer requires the new insurance contract issuer to increase its equity by some amount, but we find in our experiments that this does not change our general findings.↵

^{11}See also the empirical models presented in Bodoff and Gan [2012] and Galeotti et al. [2013]. We do not use such empirical models since the purpose of this work is to combine reinsurance and CAT bonds while the results in these articles have been obtained by focusing exclusively on CAT bonds.↵

^{12}Recall that denotes the payoff of the reinsurance contract without default risk. The actual payoff with default risk is given by Equation (7). Note also that both the reinsurance contract and the CAT bond are triggered by the insurer’s actual loss*L*, implying that basis risk is, as assumed at the outset, to be negligible._{T}↵

^{13}Although the choice of the parameters*H*and*M*is exogenous, it is not made arbitrarily in the sense that the values are drawn from the results of the problem of maximizing shareholder value. In other words, as we show later, there is in our model a demand from the insurer for such XL hedging contracts.↵

^{14}We assume that the insurer always has access to the reinsurance and CAT bond markets, regardless of the desired hedging contract, that is, there exists a supply of protection for any values of*H*_{RE},*M*_{RE},*H*_{CB}and*M*_{CB}.↵

^{15}The right hand side of Equation (20) is obtained from the relations π_{RE}= RE_{0}(1 + δ_{RE}) and π_{CB}= CB_{0}(1 + δ_{CB}).↵

^{16}As Gatzert and Kellner [2014] and others, we assume that it is too costly for the insurer to adjust its initial capital, hence considering*X*to be a constant.↵

^{17}This is not exactly true for the curve ρ = 0.7 for Panel (c) of Exhibit 4. There, we see (from right to left) that the loading initially increases as the trigger probability decreases, then reaches its maximum value and starts to decrease. This happens because the reinsurer’s and insurer’s liabilities are strongly correlated (i.e., ρ = 0.7). In this case, selling protection against extremely rare high-magnitude natural disasters may not exacerbate the reinsurer’s risk of insolvency due to the fact that the reinsurer is most likely to be already insolvent because of its own pre-existing portfolio of liabilities.↵

^{18}Although the pricing of the reinsurance contract takes moral hazard into account (see Equation (8)), in our setup, we find that it does not actually impact its price π_{RE}. This is because exposure to moral hazard is assumed to be small for the reinsurer (θ_{RE}= 0.1), and we thus generally have in Equation (8), where is the reservation price in the presence of financial distress costs (see Equation (9)). In other words, we find that the pricing of the reinsurance contract is entirely driven by financial distress costs.↵

^{19}For instance, for a volatility σ_{L}= 0.4, the ratio is 15.52% under Strategy CB, 22.38% under Strategy RE, and 32.37% under Strategy RECB. The insurer hedges a larger fraction of its risk exposure when Strategy RECB is used, because hedging is cheaper under this strategy.↵

^{20}While costs such as administration or legal fees are probably negligible for traditional reinsurance, these may play a more significant role for CAT bonds due to the SPV and the swap contract. Some hedging strategies including smaller fractions of CAT bonds may get implausible due to the relatively high expenses that occur irrespective of the expected payment and exposure.

- © 2017 Institutional Investor, Inc.