Catastrophe reinsurance provides insurance companies protection against excessive losses caused by catastrophes such as hurricanes, earthquakes, severe convective storms, flood, and winter freeze. Such protection is desired both to secure the solvency of the company as well as to stabilize underwriting results. The buyer, known as the primary insurer or ceding company, will enter into a contract with the seller, known as the reinsurer or acquiring company. Catastrophe reinsurance contracts typically come in one of three forms.

The per-occurrence treaty stipulates that the ceding company will be indemnified for losses caused by a single catastrophic event (of specified type) during the coverage period (typically one year, but sometimes a small number of years). The reimbursement is based on a formula that incorporates a retention (deductible), limit, and coreinsurance fraction. Symbolically, if \(L\) is a random variable representing the loss that the cedant incurs from a covered catastrophe, \(M\) is the limit, \(T\) is the retention, and \(r\) is the coreinsurance, then the ceded loss (amount paid by the reinsurer) \(C\) is given by \[C = r \max(0, \min(L-T, M)).\]

Per-occurrence treaties typically include reinstatement provisions. Once the limit is exhausted, it can be reinstated up to a specified number of times (typically one) by additional payments of the contract premium. Reinstatements are typically automatic. In this way, the ceding company is covered for losses due to multiple catastrophes.

Treaties often consist of multiple layers, with layers stacked one upon the other, each retention being the sum of retention and limit below it. Such multi-layer treaties are usually designed and sold through brokers; they may involve multiple sellers sharing layers or providing different layers.

Aggregate and stop-loss treaties respond to the total losses the cedant experiences over the contract period, with similar retention, limit, and coreinsurance features, but not reinstatements. In aggregate contracts, the retention and limit are expressed as fixed monetary amounts; in stop-loss contracts, they are expressed as fractions of the total premium of the covered primary insurance contracts. Because they are relatively rare, they will not be considered further here.

Further detail on the mechanics of catastrophe reinsurance can be found in Strain (1997).

Catastrophe reinsurance contracts are typically designed with retentions high enough to make payouts relatively rare, “e.g. no more often than once every five years or so” (Strain 1997). This is a practical upper limit, with most contracts we have seen designed to attach (incur first dollar payout), with 5% probability or less.

Because of the relatively low frequency of payouts, experience data is of limited information in setting a price. With the availability of commercial catastrophe models in the 1980s, pricing came to rely on model-given estimates of loss probabilities as a starting point. John A. Major and Kreps (2002) discuss how reinsurers use catastrophe models in their pricing.

Catastrophe bonds are securities (bonds) whose payment of coupons and (possibly) repayment of principle are contingent on the loss experience of an entity subject to catastrophe risk (typically a primary insurer). Catastrophe bonds thus serve a similar function to catastrophe reinsurance. Detailed explanations of their mechanics are available in Cummins and Trainar (2009) and Braun (2016).

The academic literature on the pricing of catastrophe risk is relatively limited compared to the literature on stocks and bonds. Details on particular catastrophe bonds are restricted to qualified investors. Details on reinsurance contracts are closely-held intellectual property of the buyers, sellers, and brokers. Nonetheless, some empirical studies have been conducted:

Froot and O’Connell (1999), Froot and O’Connell (2008) examine catastrophe reinsurance contracts. Rather than estimate price-risk relationships at the contract level, however, they fit industry-wide price-quantity supply and demand curves.

Morton N. Lane (2000) models catastrophe bond expected excess return (\(\mathit{EER}\), which is annual coupon minus LIBOR minus the expected loss) as a function of probability of first loss (\(\mathit{PFL}\)) and expected loss given default (\(\mathit{LGD}\)): \(\log(\mathit{EER}) = \gamma + \eta \log(\mathit{PFL}) + \beta \log(\mathit{LGD})\). Estimated parameter values are \(\eta=0.49\) (0.05 s.e.) and \(\beta=0.57\) (0.21 s.e.).

Wang (2004) fits cat bond and corporate bond rating class yield spreads to Wang-T transforms of the underlying loss probabilities. He obtains (shift, d.o.f.) parameters (0.453, 5) for cat bonds and (0.453, 6) for corporate bonds. Standard errors were not reported.

Morton N. Lane and Mahul (2008) examine models of the form \(ROL = a + b\ \mathit{EL} + c \ C(t)\) where \(C(t)\) is the value of the Guy Carpenter catastrophe reinsurance price index for year t. For their full range of data, they find b=2.052 (0.084 s.e.).

John A. Major and Kreps (2002) examine catastrophe reinsurance contracts. Their models are of the form \(\log(\mathit{ROL}) = a + b \log(\mathit{EL}) + b X\) where \(X\) is a vector built up from seven categorical predictors. They find \(b=0.53\) (0.02 s.e.).

Gatumel and Guegan (2008) examine the Morton N. Lane (2000) and Wang (2004) models, as well as another model attributed to Fermat Capital Management which is equivalent to \(ROL = EL + \lambda (\mathit{EL} (1 - \mathit{EL})/w)^{1/2}\), where \(w\) is a peril-specific weight. Their attention is focused on the shift in parameters occurring after Hurricane Katrina and they find significant shifts in the direction of higher prices.

Bodoff, Gan, and Ph (2009) model spread, defined as coupon minus LIBOR as \(a_i + b_i\ \mathit{EL}\) where the \(i\) subscripts represent levels of categorical effects including issue date, covered peril, and geographic grouping. Slope parameters \(b_i\) range from 1.48 to 2.49.

Galeotti, Gürtler, and Winkelvos (2013) cover linear, log-linear, and Wang transform model specifications. Their objective is to evaluate the various model forms. They also examine covariate predictors including type of trigger, covered perils, bond rating and maturity, and various external indices. In the linear model without covariates, the EL parameter was estimated at 2.42 (s.e. 0.10).

Gürtler, Hibbeln, and Winkelvos (2016) use a linear-EL specification with numerous covariates including peril, region, rating, volume, various time effects, bond random effects, and macroeconomic variables. In total, 17 models are reported with EL coefficients ranging from 1.257 to 3.058 (median 2.584).

Braun (2016) also uses a linear-EL specification with 17 additional main and interaction effects. They report 11 model results with EL coefficients ranging from 1.947 to 3.137. They also re-fit the Morton N. Lane (2000), John A. Major and Kreps (2002), and Gatumel and Guegan (2008) models as well as fitting a quadratic in log(EL) and compare their preferred model to the others in terms of out-of-sample performance.

John A. Major (2019) provides a detailed analysis of different methodological considerations for statistical models of cat bond prices.

- How does the reinsurer price
- What is insurer’s optimization problem
- What is set of available contracts (e.g. optimize over XOL and determine attachment, attachment and participation, over all increasing functions, continuous etc. )
- Assumptions about insurers loss distribution (e.g. continuous, mass at zero)
- Some add a budget constraint

*Expected value premium principle* is consistently used for re pricing via \((1+\rho)\mathsf{E}[C]\) where \(C\) is ceded loss and \(\rho\ge 0\) is a safety loading. Some use SRM for re pricing, which does not make sense IMO.

Optimize risk of net plus cost of re. Risk is VaR, CVaR or SRM applied to net.

Indemnity functions: net and ceded increase with loss quite common. Specification critical. E.g. can restrict to determine attachment by restricting set of indemnities.

Mostly only looking at whole account, aggregate covers.

Mostly saying the net cost is VaR or CVaR—huge opportunity

VaR can lead to non-increasing optimal contracts - don’t buy for the really big events.

Min risk of net plus cost of re, EV for re, over attachment by line in per occ. Unlimited protection. Focus on risk sharing.

Numbers in parenthesis give the number of Google Scholar citations.

Borch (1962) (1128) Econometrica, Equilibrium in a Reinsurance Market.

- risk sharing between insurers with utility functions.
- Additive pricing - hence function of cumulants. All cos buyers and sellers.

N. A. Doherty and Tinic (1981) (167) JofF Reinsurance under conditions of capital market equilibrium: a note

- “The purpose of this note is to establish the effect of reinsurance on the insurer’s common stock when it is traded in an efficient capital market.” Shareholders can diversify themselves.
- “reinsurance can increase the value of the insurance company when premium income is not independent of the risk of insurance company ruin”

Young (1999) IME Optimal insurance under Wang’s premium principle, with utility

- Not re specifically but optimal contracts under Wang pricing
- X has mass at zero but continuous above zero
- Derives properties of optimal ins
- Excess with a ded, but utility-dependent excess function (Thm 3.6)

Gajek and Zagrodny (2004) (101) IME, Optimal reinsurance under general risk measures.

- One sided harm functions drive insurer decision
- Admissible contracts: less than loss; between two other functions of gross
- Re: standard deviation pricing rule
- Consider abs dev risk measure \(L^1\) risk measure
- Gives sufficient conditions for optimality
- Under abs dev risk measure: XOL layer (Thm 3.1, gives formula for limit and attachment which balances loss below and above program) or share of unlimited XOL shown optimal (Thm 4.1 truncated variance risk measure)

Kaluszka (2005) (89) IME, Optimal reinsurance under convex principles of premium calculation

N. Doherty and Smetters (2005) (105) JRI, Moral hazard in reinsurance markets.

Plantin (2006) (72) JRI, Does reinsurance need reinsurers?

- Pooling model not good…risk is not pooled in the expected way, it is ceded in a pyramid structure to professional res. Sees role of res to monitor primary insurers credibly so they can raise capital more easily (
*tighter g*). Costly capital.**Quotes model like Holmstrom and Tirole (1997) (4043) QJE Financial Intermediation, Loanable Funds, and the Real Sector.**

Cai and Tan (2007) (242) ASTIN, Optimal Retention for a Stop-loss Reinsurance Under the VaR and CTE Risk Measures.

- Only stop loss reinsurance, only deciding retention
- Re price safety loading above expected
- Opt risk(net) + cost of re
- Explicit conditions and formulas for optimal retentions.
- No partial placement
- Only WASLs

Cai et al. (2008) (238) IME, Optimal reinsurance under VaR and CTE risk measures

- Minimize risk of net plus cost of re
- Quota share, stop loss, change-loss = partially placed stop loss
- increasing convex cession function; approx with stop losses at different attachments
- re priced at expected plus safety loading
- Thm 3.1 find above functions as optimal depending on VaR threshold, X and safety loading

Bernard and Tian (2009) (113) JRI, Optimal reinsurance arrangements under tail risk measures

Balbás, Balbás, and Heras (2009) (123) IME, Optimal reinsurance with general risk measures

Ka Chun Cheung (2010) (109) ASTIN, Optimal Reinsurance Revisited - A Geometric Approach.

- Examine Cai et al. (2008), find simpler and more transparent solution.
- Apply method when re priced with Wang (Thm 3) minimize Var Net + cost re (implies indemnity functions are comon with gross losses, bottom of p. 235). Comes up with solutions full cession or partial QS. No excess! Proof only seems to work with partially placed excesses and shows attach=0. Proves optimal ceded loss functions must be partial XOL layers.
- Quite nice.

Chi and Tan (2011) (128) ASTIN Optimal Reinsurance under VaR and CVaR risk measures: a simplified approach

- Studies to date identify stop loss, limited stop loss \((x-a)^+\wedge b\), or truncated stop loss \(f(x)=(x-d)^+1_{\{x \le m \}}\) as optimal.
- Consider classes of indemnity functions: increasing convex, ceded and retained are increasing, retained losses are left continuous and increasing. Under VaR optimal contracts are stop loss, limited stop loss and truncated stop loss respc. Under CVaR stop loss is always optimal.
- Both ceded and retained increasing manages moral hazard.
- Re pricing: expected loss plus a safety loading
- Optimization problem: minimize VaR or CVaR of net loss plus cost of re = risk of net plus cost of re.
- Shows increasing convex are mixtures of stop loss contracts
- Identifies optimal retentions etc. over each class of functions.
- Uses f(Var)=Var(f) for nice functions f
- Method of proof: show there is no better function than the explicitly displayed one (e.g. Prop 3.1)

Hürlimann (2011) (21) ASTIN, Optimal reinsurance revisited - Point of view of cedent and reinsurer.

Tan, Weng, and Zhang (2011) (77) IME, Optimality of general reinsurance contracts under CTE risk measure

- price with expected plus risk load
- min CVaR of net plus cost of re (2.8)
- optimize re buy with budget
- Comes up with optimal buys that are not comon with gross, fig 2
- stop loss is an optimal solution but “more elaborate” functions can also be optimal
- sometimes optimal to spend less than budget

Bernard (2013) (15, survey): Risk Sharing and Pricing in the Reinsurance Market

Cui, Yang, and Wu (2013) (69), IME, Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles

**ABSTRACT** This paper discusses the optimal reinsurance problem with the insurer’s risk measured by distortion risk measure and the reinsurance premium calculated by a general principle including expected premium principle and Wang’s premium principle as its special cases. Explicit solutions of the optimal reinsurance strategy are obtained under the assumption that both the ceded loss and the retained loss are increasing with the initial loss.

- Distortion to measure ins risk
- re priced with a different distortion
- ceded loss functions: ceded and net increasing functions of gross
- optimize with and without a budget constraint
- Thm 2.1: come up with the minimum which looks at when ins distortion is below, equal or above the re’s. …complex but looks like a bunch of layers. See fig 2.1 stair-step cession function
- Nice, but looks complicated.

Chi and Tan (2013) (87) IME,Optimal reinsurance with general premium principles

**ABSTRACT** In this paper, we study two classes of optimal reinsurance models from the perspective of an insurer by minimizing its total risk exposure under the criteria of value at risk (VaR) and conditional value at risk (CVaR), assuming that the reinsurance premium principles satisfy three basic axioms: distribution invariance, risk loading and stop-loss ordering preserving. The proposed class of premium principles is quite general in the sense that it encompasses eight of the eleven commonly used premium principles listed in Young (2004). Under the additional assumption that both the insurer and reinsurer are obligated to pay more for larger loss, we show that layer reinsurance is quite robust in the sense that it is always optimal over our assumed risk measures and the prescribed premium principles. We further use the Wang’sandDutchpremium principles to illustrate the applicability of our results by deriving explicitly the optimal parameters of the layer reinsurance. These two premium principles are chosen since in addition to satisfying the above three axioms, they exhibit increasing relative risk loading, a desirable property that is consistent with the market convention on reinsurance pricing.

- net and ceded increase with gross
- re pricing satisfies
- LI
- risk loading: prem is \(\ge\) ceded loss
- stop loss order preserving (which seems reasonable but is actually BS)

- min C/VaR of net loss plus re premium
- Given re policy \(f\) make \(h_f\) layer with deductible \(\mathsf{VaR}_p(X)-f(\mathsf{VaR}_p(X))\) and limit \(f(\mathsf{VaR}_p(X))\). Note cession at VaR is the same viz \(f(\mathsf{VaR}_p(X))\)
- Thm 3.1: substituting \(h_f\) for \(f\) is better. Similar for CVaR (3.2)
- Give solution for Wang (4.2) and Dutch prem princ for C/VaR

Tan and Weng (2014) (23), NAAJ, Empirical Approach for Optimal Reinsurance Design

K. C. Cheung et al. (2014) (68) Scand, Optimal reinsurance under general law-invariant risk measures

Zheng, Cui, and Yang (2014) (15) J. Syst. Sci and Complexity, Optimal reinsurance under distortion risk measures and expected value premium principle for reinsurer

- re expected plus safety loading
- retained loss increasing with gross
- insurer risk is SRM applied to net plus cost of re
- explicit solution in terms of g, with indemnity combo of truncated stop losses
- strangely do not restrict to increasing concave \(g\) see Assumption A, p. 125
- get some whacky looking indemnity schedules, Fig 2
- REVISIT

Zheng and Cui (2014) (26), IME Optimal reinsurance with premium constraint under distortion risk measures

Recently distortion risk measure has been an interesting tool for the insurer to reflect its attitude toward risk when forming the optimal reinsurance strategy. Under the distortion risk measure, this paper discusses the reinsurance design with unbinding premium constraint and the ceded loss function in a general feasible region which requiring the retained loss function to be increasing and left-continuous. Explicit solution of the optimal reinsurance strategy is obtained by introducing a premium-adjustment function. Our result has the form oflayer reinsurance with the mixture ofnormal reinsurance strategies in each layer. Finally, to illustrate the applicability ofour results, we derive the optimal reinsurance solutions with premium constraint under two special distortion risk measures—VaR and TVaR.

- Extends Zheng, Cui, and Yang (2014)
- includes premium constraint

Chen, Wang, and Ming (2016) (1) Risks, Optimal Reinsurance Under General Law-Invariant Convex Risk Measure and TVaR Premium Principle

Cong and Tan (2016) (10) Annals OR, Optimal VaR-based risk management with reinsurance

Chi et al. (2017) (7) NAAJ, Optimal Reinsurance Under the Risk-Adjusted Value of an Insurer’s Liability and an Economic Reinsurance Premium Principle.

- minimize net risk adjusted value using C/VaR capital and cost of capital approach
- net and ceded increase with gross loss
- re uses econ prem: \(\mathsf{E}[XM]\), where \(M\) is market factor \(w(Z)/\mathsf{E}[w(Z)]\). Very general. NOT LI.
- Come up with surprisingly simple optimal res, Thm 3.1 and 3.2.
- Actually looks quite nice, but takes us too far afield.

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Borch, Karl. 1962. “Equilibrium in a Reinsurance Market.” *Econometrica* 30 (3): 424–44.

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