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Wanting robustness in insurance: A model of catastrophe risk pricing and its empirical test
Accepted Manuscript Wanting robustness in insurance: A model of catastrophe risk pricing and its empirical test Wenge Zhu
PII: DOI: Reference:
S0167-6687(17)30045-8 http://dx.doi.org/10.1016/j.insmatheco.2017.08.006 INSUMA 2378
To appear in:
Insurance: Mathematics and Economics
Received date : January 2017 Revised date : August 2017 Accepted date : 19 August 2017 Please cite this article as: Zhu W., Wanting robustness in insurance: A model of catastrophe risk pricing and its empirical test. Insurance: Mathematics and Economics (2017), http://dx.doi.org/10.1016/j.insmatheco.2017.08.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Wanting robustness in insurance: A model of catastrophe risk pricing and its empirical test Wenge ZHU* School of Finance Shanghai University of Finance and Economics Shanghai, 200433, P. R. China
Abstract: Motivated by the fact that a lack of information about natural disasters may lead agents to be ambiguity averse to catastrophe risks, we introduce a new type of penalty function and propose an adjusted equilibrium model based on the function by allowing agents to act in a robust control framework against model misspecification with respect to rare events. The pricing formulas are then derived for various catastrophe linked securities such as catastrophe futures, options and bonds. We also estimate and test the model using empirical data of catastrophe bonds and compare it with various other models and investigate the robustness performance of alternative pricing formulas. Key Words: Ambiguity Aversion; Robust Control Theory; Catastrophe Risk Pricing JEL Classification: G12 G13 G28
* Tel: +86-21-65908401; e-mail: [email protected]
1. Introduction Economic losses from natural catastrophes such as hurricanes, earthquakes, and floods have increased significantly in China as well as in other parts of the world in recent years. There are two main factors driving the increasing losses from natural disasters: degree of urbanization and value at risk. These have not only generated greater insurance exposure levels but also rendered past economic and insured losses less relevant when attempting to estimate future scenarios. Usually losses from natural disasters are financed using one of five mechanisms: private insurance, capital market securities, government assistance, self-insurance, and charity. Although private insurance system is widely regarded as a more efficient mechanism for financing and managing the pure risk, it seems that insurance coverage is extremely limited for catastrophe losses and many exposures faced by corporate and household sectors never even reach insurers. For example, the 2008 Wenchuan Earthquake in China resulted in an estimated RMB 1.6 billion in insured claims, or only 0.2% of the total losses caused by the earthquake, while property insurers even in the United States have only provided an estimated $57 billion or 32% of total losses for the Hurricanes Katrina, Rita and Wilma in 2005. The paucity of insurance funding for large event risk has been considered as a barrier to better sharing of catastrophe risk and there are a lot of discussions among practitioners, academics, and policymakers on why there is a paucity of catastrophe risk sharing by the insurance sector, what are the weaknesses of the current system of financing disaster risk, and what kinds of alternative methods may be created to help solve the problem of inadequate catastrophe risk sharing (see., e.g., Froot, 1999 and Cummins and Barrieu, 2013, for reviews of such discussions). One of the most significant innovative catastrophe risk financing techniques has been through new financial instruments like CAT futures, options and bonds that bring catastrophe exposures directly to the capital market. One major driver of nonlife securitization is the so-called insurance underwriting cycle, which means that catastrophe insurance and reinsurance market usually undergoes alternating periods of soft markets, when prices are relatively low and coverage is readily available, and hard markets, when prices are high and coverage supply is restricted (Harrington et al, 2013). The previous literature has provided some explanations for the observed cycle patterns in the insurance market. For example, Froot (1999) examines eight potential explanations and concludes that the most compelling explanation is that holding capital is costly for insurers and reinsurers because of informational asymmetries between firms and the capital market. It is thus expected that the catastrophe securitization which provides risk exposure direct access to capital market has the potential to moderate the effects of the insurance underwriting cycle. Among all catastrophe financial products, it seems insurance-linked CAT bonds are by far the most successful securitized risk hedging instrument. Nevertheless, the CAT bonds market is still not immune to the underwriting cycle and the cyclical behavior of the CAT bonds market is very similar to a “typical” reinsurance market during the alternating periods of soft markets and hard markets. The cyclical behavior 1
of CAT bond prices seem to be puzzling in view of the fact that capital markets are not exposed to the imperfections of insurance markets. Zhu (2011) pointed out two other stylized facts of CAT bond prices that seem contradictory to traditional capital market theories. One is that the spread premium of “zero beta” CAT-linked bonds seems very high relative to the expected principal loss, the other is that spread premium is more pronounced for CAT securities with lower expected loss ratios thus generate a kind of “smirk” pattern in the cross-sectional plot of the premium to E[loss] ratio against the expected loss ratios. It is thus interesting to explain the facts by extensive theoretical economic analysis. The academic research on the role of insurance sector usually starts by examining a perfect economic model initially developed by Arrow (1953), with a specific extension to insurance market by Borch (1962) and Bühlmann (1980). The efficiency of the Arrow-Borch model for insurance risk financing depends on the following conditions: 1. There are a large number of agents in the economy and agents are expected utility maximizers, each with an increasing and concave utility function. 2. There is no transaction cost. 3. Either all agents face similar risks, or differences in risk can be observed at no cost and be incorporated in insurance premiums. 4. Insurance does not dissuade those who are insured from avoiding risks. The prevention efforts of agents in the economy are observable at no cost. 5. The corporate form for insurers is efficient. 6. The insured losses can be modeled with precision. In many insurance markets, one or several of the six conditions may not hold. Violations of the first and sixth condition are particular features of catastrophe risk market. While model misspecification is already a big problem for non-catastrophe risks which are extensively modeled, the catastrophe risk modeling is greatly hamstrung by the paucity of quality data, and this is further aggravated by lack of catastrophe models for different types of natural disasters in different parts of the world. Limitations on data and assumptions about a catastrophe model affect the model estimates and thus there is significant uncertainty or ambiguity associated with these estimates. Controlled experiments with underwriters and actuaries reveal that insurers would violate the axioms of expected utility theory and charge more premiums, in other words, insurers appear to be ambiguity averse if there is considerable uncertainty with respect to risk (Cabantous et al, 2011). Ambiguity aversion is not a new concept in economic theory and may be traced back to Knight (1921) and Keynes (1921). The basic point is that while a probability measure can represent likelihood assessment, it cannot also model the other dimensions of beliefs such as confidence of the likelihood assessment. Since it is impossible to distinguish the so-called ‘Knightian uncertainty’ or ‘ambiguity’ from ‘risk’ within the traditional subjective expected utility framework, it becomes important to formulate a new theory that permits a distinction between risk aversion and ambiguity aversion. In recent years, there have been many researches in financial economics literature 2
recognizing the presence of ambiguity and illustrating its potential role in explaining several asset market empirical puzzles (see, e.g., Chen and Epstein (2002), Klibanoff et al (2005), Liu et al (2005) and Ju and Miao (2012)). Ambiguity aversion is also not new to insurance and actuarial professionals. In fact, it may be categorized into the credibility theory in actuarial science which provides a framework examining the effect of learning the imprecise knowledge about insurance fundamentals by successive approximations (Bühlmann 1970). See, for example, Cairns (2000) for an exposition of the problem concerning both parameter and model uncertainty in the traditional actuarial learning approach. But just as Savage’s decision theory is inconsistent with the Ellsberg paradox evidence of ambiguity aversion, the traditional actuarial learning approach also raises two kinds of concern about model misspecification: one is that the chosen models set might be wrong or that some other fitting models might be better; the other is that even when we can specify the set of alternative models, we might have doubt how to choose and estimate a model from the set to fit the data. Zhu (2011) adopted the so-called robust control framework introduced in Hansen and Sargent (2001) and takes the choice patterns in the Ellsberg paradox or ambiguity aversion into consideration to explain the above-mentioned three stylized facts concerning catastrophe linked securities pricing. In the robust control settings, the perspective of a decision maker differs substantially from that in the traditional learning theory. The agent is assumed to deal with model uncertainty as follows: First, having noticed the unreliable aspects of the reference model based on existing information, he evaluates the future prospects under alternative models. Second, acknowledging the fact that the reference model is indeed the best statistical characterization of the available information, he penalizes the choice of alternative model by a distance function measuring how far it deviates from the reference model. The robust control theory uses the concept of relative entropy as a distance function to define a kind of objective function of the robust decision-maker with the relative entropy as a penalty term measuring the ambiguity aversion magnitude of the agent. But Zhu (2011) finds that the ambiguity aversion premium emerging from the relative entropy penalty function choice in Hansen and Sargent (2001) seems too high for the purpose of CAT risk pricing, so he introduces a more general penalty function form which may be chosen to fit empirical test. However Zhu was unable to give an explicit form of the function. So in this paper we will introduce a new concrete explicit form of the penalty function and propose an adjusted equilibrium model based on the function by allowing agents to act in a robust control framework against model misspecification with respect to rare events. The rest of the paper is organized as follows. Section 2 sets up the equilibrium model by introducing the new penalty function in the robust control framework. Section 3 demonstrates how to use the equivalent martingale measure derived in section 2 to value exchange-traded catastrophe securities and CAT bonds. Section 4 presents estimation and robustness test of the model using empirical CAT bonds data and gives a further discussion comparing our model with several previous related models. Section 5 concludes. 3
2. Robust Control Theory and Economic Model We first formulate our modeling framework of robust control theory and assume there is a catastrophe risk market where catastrophe loss Yt is traded and the market exchanges bundles of state-contingent contracts that can be analyzed as catastrophe-linked securities. The aggregate CAT losses Yt is assumed to follow a compound Poisson process; that is, dYt = Lt dN t ,
(2.1)
where Nt is a Poisson process with intensity λ>0 and Lt is the conditional random loss amount at time t, which is independent of Nt. For convenience we assume Lt is described by identical distribution function, Pr( Lt x ) FL ( x ) where FL(x) denotes the distribution function of a random variable L. A representative agent starts with an initial wealth w at time 0 and we further assume that the agent receive a premium of mc at that time to underwrite a part of CAT risk, where c is the price of the insurance covering total CAT losses between time 0 and a terminal, prespecified time T; m denotes the proportion the agent underwrites in the full insurance (we assume the proportion does not vary between time 0 and T for convenience here but the insurance proportion can also be allowed to adjust during the time interval). We assume all loss payments occur at time T and the continuously compounded risk-free interest rate r is a constant. Hansen and Sargent (2001) measure discrepancy between alternative model and reference model by ‘relative entropy’, defined as the expected value of a log-likelihood ratio (the log of the Radon-Nikodym derivative). More exactly, letting P=(Pt) be the probability measure associated with the reference model, the alternative ~ ~ ~ model be described by a probability measure P ( Pt ) with t dPt / dPt as the ~ Radon-Nikodym derivative of Pt with respect to Pt, the relative entropy is then
1~ E t [log( t )] , where the conditional expectation is evaluated 0 t
defined by I t lim
with respect to the density associated with the alternative model. Since it seems that the uncertainty aversion in practice only applies to the likelihood component of the CAT arrival, we effectively assume the agent in our model only has doubt about the probability of CAT events and the relative Radon-Nikodym derivative t is defined by the following stochastic differential equation:
dξt (e ht 1)ξt dNt (eht 1) λξt dt ,
(2.2)
where ht is a time dependent positive function controlling the model distortion magnitude, and 0 1 . By the construction, the process 4
{t , 0 t T } is a
~ ~ martingale of mean one. The measure P ( Pt ) thus defined is indeed a probability
measure, and we later denote P as the entire collection of such probability measures. Given the alternative model specification defined by (2.2), the distance measure It can be derived as: h h It= (e t ht e t 1) .
(2.3)
Using a penalty function of the relative entropy It, Zhu (2011) defines a kind of objective function of the robust decision-maker with the penalty function as a term measuring the ambiguity aversion of the agent. More exactly, in this generalized robust control framework, if we do not consider the randomness of consumption during the period, the agent’s time-t objective function is defined recursively by T
~ U(t)=U(W, t, y, m)= inf ~ {E t [u (WT ) (U ( s ), I s , s )ds ] Wt W , Yt y} , P P
(2.4)
t
with boundary condition U (W , T , y , m) u(W my ) and for convenience in the later paragraphs we further suppose u( x ) (1 / )e x , in which >0 is the risk aversion coefficient; WT
denotes the accumulated wealth of the agent at time T;
T
(U ( s), I , s)ds is s
a function of U and the distance measure Is that controls the
t
ambiguity aversion magnitude of the agent. More precisely, the penalty term in Zhu (2011) at time s is defined as:
(U ( s ), I s , s ) U ( s ) g s ( I s ) ,
(2.5)
where g s is assumed to be an increasing and convex function with g s (0) 0 so that T
T
t
t
(U ( s), I s , s)ds] = U ( s) g s ( I s )ds is indeed a penalty term. Zhu was unable to give an explicit form of the function but we find that if we define
g ( I s ) g ( (e hs hs ehs 1))
(ehs 1) 1 , ( 1)
(2.6)
in which both and are positive parameters, with associated with the general ambiguity aversion magnitude and the term associated with controls the distance function (eht ht eht 1) . It can be proved that the penalty form thus defined indeed 5
satisfies the increasing and convex function requirement and we later show it can exactly fit the empirical test for catastrophe risk pricing and discuss more about its possible economic meaning in the conclusion. With the above explicit form of the penalty term, the definition of the objective function can be formulated as a random optimization problem and satisfies the following HJB equation
Ut rWUW inf {e ht [EU (W , t , y L, m) U (W , t , y, m)] λγ ht
(e ht 1) β 1 U } =0, (2.7) α( β 1)
where Ut is the derivative of U with respect to t and UW is its first derivative with respect to W. With the above proper form assumption of the objective function, the first order condition for ht gives a constant solution given by ht h* ln(1 [ (
M L (m) 1
)]1 / ) ,
with ML denotes the moment generating function of the loss random variable L. (see, e.g., Zhu, 2011) Substituting the objective function into the HJB equation and the solution of Equation (2.7) at time 0 with W=w+mc, y=0 is given by
1
T
U ( w mc,0,0, m) exp{ ( w mc)e [e ( M L (m) 1) rT
h*
0
( e h* 1)
]dt} .
The first order condition for m (the second order condition also holds) in the above equation gives the following equation for the optimal insurance proportion m*: T
c e rT e h * E[ Lem* L ]dt e rT e h *E[ Lem* L ]T 0
(2.8)
Since in equilibrium the market is cleared and the representative agent will accept total catastrophe loss under the equilibrium price, in other words in equilibrium we should have m*=1, the solution to the market equilibrium and the corresponding EMM can then be summarized by the following proposition: Proposition 1: In equilibrium and under the assumptions above, the price of CAT risk is given by c e rT e h E[ LeL ]T ,
(2.9)
with
h ln(1 [ (
M L ( ) 1
)]1 / ) .
(2.10)
The pricing formula (2.9) can also be rewritten as an expectation with respect to an equivalent martingale measure (EMM) Q in the following equivalent form: c E Q0 [ e rT Y T ] E [ e rT Y T
exp( Y T hN T ) ] , E [exp( Y T hN T )]
(2.11)
with the Radon-Nikodym derivative of Q with respect to P given by dQ/dP 6
exp(YT hN T ) / E[exp(YT hN T )] .
Proof: Proof of Formula (2.9) is direct. See Appendix B of Zhu (2011) for the proof of the equivalent martingale measure expectation expression of (2.11). Now we apply the pricing kernel derived above to the pricing of more general catastrophe-linked securities. These kinds of securities may be looked as derivatives contingent on catastrophe losses. Let φ be a measurable function that specifies the payoff at maturity to the buyer of the catastrophe-linked securities, i.e., at time T the buyer receives φ(YT). By augmenting the new type of contingent claim contract in the above equilibrium model, it can be proved that the price of a contingent claim at time 0 with payoff φ(YT ) at time T is given by c E Q0 [ e rT (Y T )] E [ e rT (Y T )
exp( Y T hN T ) ]. E [exp( Y T hN T )]
(2.12)
Remark 1: We assume the insurance price is determined so that the agent underwrites the full insurance at time 0. The price may change in the later time to assure the agent’s total underwriting of the catastrophe risk. The difference of the EMM derived here from the traditional ones is that we have augmented an exponential of hNT to represent the ambiguity aversion of the agent (h depends on and according to Equation 2.10). Notice that h also depends on the distribution function of the random variable L, so if the distribution assumption of L varies in the future, we will induce a fluctuation of the spread premium coming from the ambiguity aversion thus the cyclical insurance price behavior may emerge endogenously from our model. Another way to make insurance price varies is to include learning into the robust control model, which Hansen and Sargent (2010) describes as robust learning theory. We will discuss more about it in the conclusion. Remark 2: If h becomes zero, i.e., if the market shows no ambiguity aversion, the form of dQ/dP then reduces to the traditional Esscher transform, which has been used for a long time in insurance and finance pricing (Bühlmann, 1980; Gerber and Shiu, 1994 and Yang, 2006). Wang (2004) shows that a generalized two-factor Wang Transform may be used for the pricing of CAT bonds by capturing economic behavior of greed and fear as well as risk aversion. It will be interesting to study whether the two-factor Wang Transform can be related with and grounded in our robust control economic framework. Remark 3: Epstein and Schneider (2003) criticized that robust control theory may cause dynamic consistency problem and propose a remedy to expand the alternative models set to attain what they called rectangular condition. But Hansen and Sargent (2016) pointed out that such an expansion may allow too much freedom in setting the alternative models for the use of a meaningful robust control problem. Since we have 7
assumed the conditional loss amount Lt as described by identical distribution function, the dynamic consistency problem may be solved by requiring ht as a constant in our alternative model specification defined by (2.2). 3. Catastrophe-linked Securities Pricing 3.1. Exchange-traded Catastrophe-linked Derivatives Pricing In this section we first apply the pricing kernel derived in the previous section to the pricing of exchange-traded catastrophe-linked securities. Hurricane Andrew in 1992 raised capital shock problem of the insurance and reinsurance industries and as a result, market participants began to explore alternative securitization measures for hedging catastrophic risk. The first such effort was the introduction of catastrophe futures and options by the CBOT in 1992. The 1992 contracts were replaced in 1995 with redesigned catastrophe call spread based on loss indexes provided by PCS, an independent insurance agency, but the CBOT-PCS contracts were delisted in 2000 due to lack of investor interest. There have been several recent efforts to relaunch securitization with payoffs triggered by catastrophic property losses. In 2007, futures and options contracts were introduced by the New York Mercantile Exchange (NYMEX), the Chicago Mercantile Exchange (CME), and the Insurance Futures Exchange (IFEX), whose contracts trade on the Chicago Climate Exchange (CCX). The IFEX futures are similar to the CBOT ones in using PCS index triggers but their contract payoff type is binary, while CME contracts are similar to the CBOT contracts in that they are not binary but are valued by a parametric index. Both types of contracts are subject to substantial basis risk and since it seems the problems with the failure of the CBOT contracts still have not been overcome with respect to the IFEX and CME products, it still remains unclear whether the new catastrophe derivatives market can succeed (Cummins and Barrieu, 2013). Since the contract structures are similar for the CBOT products with the new ones, we will only focus on CBOT contracts to illustrate the pricing of exchange-traded catastrophe-linked securities. Following Aase (1999), we make some simplifying assumptions that the catastrophe risk process Yt is a compound Poisson process with the distribution function of the claim size amounts L as identical Gamma distributed and then derive the explicit formulas for some specific forms of exchange-traded catastrophe -linked derivatives. Let the gamma distribution function be given by,
( x; a, b)
b a x a 1 by y e dy, x 0 . (a) 0
The moment generating function of the corresponding YT then becomes
M Y ( z, T ) E[exp(zYT )] exp[T [(
b a ) 1]] . bz
Hence the corresponding moment generating function with the EMM Q in Proposition 1 becomes 8
M Y ( z, T ; , h ) EQ [exp( zYT )] exp{Teh (
b a b a ) [( ) 1]} , b b z
(3.1)
which shows that the transformed process is of the same type, with parameters
( , a, b) replaced by (eh (
b a ) , a, b ) . b
Example 1: Catastrophe Futures Contract The first contract we consider is the catastrophe futures. We abstract from most details of the CBOT contracts and focus on the key mathematical features that enter into this type of contract. Define an insurance loss ratio index by the insured catastrophic property loss divided by , which is usually one-fourth the dollar value of the premium collected in the previous year. By CAT futures contract we mean a financial instrument with a stated value of $25,000 multiplied by the loss ratio index, i.e., the terminal payoff function φ(x) at time T can thus be described as
( x ) $25,000(
x F) ,
with F denotes the futures price determined at time 0. We then have the following futures pricing formula. Proposition 2: Under the assumption above, the following futures formula at time 0 is
F
eh ba 1aT . ( b ) a 1
(3.2)
Proof: From the above discussion about compound Poisson process with Gamma distributed claim size amount, YT under the equivalent martingale measure Q is also a compound Poisson process with Gamma distributed claim size. For the futures contract, we should have
0 E0Q [25,000(
YT F )] ,
by the formula (2.12), which implies that
F E0Q [
ehba 1aT 1 YT ] EQ0 [YT ] . ( b ) a 1
Example 2: Catastrophe Options Contract The second contract we consider is the catastrophe options. By this we mean a financial instrument with terminal payoff function φ(x) at time T described as
( x) $25,000(
x K )
with K denotes the strike price of the option. We then have the following pricing 9
formula for catastrophe options contract. Proposition 3: Under the assumption above, the following options pricing formula at time 0 is
(K ) e rT 25,000e
e h (
b a ) T b
( e h (
j 1
(
b a j ) ) b j!
ka [1 ( K , ja 1, (b )] K [1 ( K , ja, (b )]) (b )
(3.3)
Proof: Similar to the derivation in Proposition 2, the price of catastrophe options is given by
( K ) E0Q [e rT 25,000(
e rT 25,000e
e h (
YT K ) ]
b a ) T b
( e h (
j 1
where B(j) denotes the expected value of (
b a j ) ) b B( j ) , j!
L1 L2 ... L j
K ) under the adjusted
Gamma distribution, which can be calculated as
B( j )
ja [1 ( K , ja 1, (b )] K [1 ( K , ja, (b )] , (b )
and by this we obtain the conclusion of the proposition. Example 3: Catastrophe Spreads Contract In 1995 CBOT catastrophe futures and options contracts were replaced by catastrophe call spreads based on loss indexes provided by PCS. Again we abstract from the CBOT-PCS contract details and focus on its mathematical features. For this contract, the terminal payoff function can be described as:
( x) $25,000[(
x x K1 ) ( K 2 ) ]
where K1 denotes the lower strike price and K2 denotes the upper strike price. By capping of the options the spreads contract will limit the risk of its investors. The market price of the contract will then be given by the following proposition. Proposition 4: Under the assumption above, the following spread price formula at time 0 is
( K1 ) ( K 2 ) ,
(3.4) 10
with ( K1 ) and ( K 2 ) as given by the formula in Proposition 3. Proof: It can be directly derived from the payoff function form of the spreads contract. 3.2 Catastrophe Bonds Pricing We now apply formula (2.11) to catastrophe bonds pricing and do not need Gamma distributed claim size assumptions any more. CAT bonds market started during the 1990s and has matured and become a steady source of capacity for insurers and reinsurers since then. The CAT bonds market grew from less than $1 billion per year in 1997 to $7 billion in 2007. Even though the market was affected by the subprime financial crisis of 2008, CAT bonds issuance in 2014 already exceeded 2007, which previously was the largest year on record (Swiss Re, 2016). CAT bonds are usually priced at spreads over LIBOR, meaning that investors receive floating interest plus a spread or premium over the floating rate. We assume for convenience of discussion that the form of CAT bonds is described as follows: The CAT bond is priced at p, which denotes the bond principal. If the loss for a single CAT event in the period (0, 1) is less than loss trigger B1, the agent will get ~ back his principal p, and spread premium plus interest p(er 1 l ) at maturity time ~ 1, in which r is the LIBOR and l denotes the spread premium rate. Once a CAT loss
exceeds the trigger, the agent will forfeit some or all the principal at time 1. The loss fraction f12(L) of the principal is in proportion to the conditional loss L in the range between the trigger B1 and a cap B2 and is given by f12 ( L ) Max[0, Min ( L B1 , B2 B1 )] /( B2 B1 ) . We assume the spread premium plus
the risk-free interest is guaranteed no matter whether any principal loss occurs. The expected value of the bond principal loss proportion is denoted as l. ~ The ratio of the spread premium l to the expected principal loss ratio l is then given by the following proposition. Proposition 5: Under above assumptions, the ratio of the spread premium to the expected principal loss proportion (i.e., the ratio of the modified expected principal loss to the expected loss) of the above CAT bond is given by h ~ l (1 e e M L ( ) ) E[ f12 ( L )eL ] , (3.5) l (1 e )E[ f12 ( L )]M L ( ) with h given by (2.10). Proof: The loss fraction f12 ( L ) of the principal is given by f12 ( L ) Max[0, Min ( L B1 , B2 B1 )] /( B2 B1 ) 11
(( L B1 ) ( L B2 ) ) /( B2 B1 ) .
(3.6)
Thus the cash flow the agent gets back at time 1 can be described as ~ (Y1 ) p (er l I ( N1 0) f12 ( L)) , where I is an event indicator function, L denotes the first conditional catastrophe loss. The probability that at least one catastrophe loss occurs in the period is given by E[ I ( N1 0)] (1 e ) . The mean of the bond principal loss proportion is thus given by l E[ I ( N1 0) f12 ( L )] (1 e ) E[ f12 ( L)] . Now we apply the equivalent martingale measure Q in Proposition 1 instead to price the above CAT bond. The basic equation can be written as: ~ p EQ0 [ p e r ( e r l I ( N1 0) f12 ( L))] , from which we can get the formula to calculate the spread premium as below, ~ l E[exp(Y1 hN1 ) I ( N1 0) f12 ( L)] / E[exp(Y1 hN1 )] , which can be calculated as ~ (1 e e l
h
M L ( )
)E[ f12 ( L )eL ] , M L ( )
(3.7)
thus the spread premium to the mean principal loss ratio is as given in (3.5). From Equation (3.7), we can find that the spread premium emerging from risk aversion is relatively small and is not important for CAT bonds pricing (see, e.g., Zhu, 2011). So in later paragraphs, we will only focus on the effect of ambiguity aversion and assume the market is risk neutral to the catastrophe risk, in other words, we will assume 0 later. Under this assumption, the equation (3.5) then reduces to h ~ l (1 e e ) E[ f12 ( L )] 1 [E( L)]1 / . l (1 e )E[ f12 ( L )]
(3.8)
Given the lack of empirical data for E[L], we are unable to directly apply the formula (3.8) and thus are forced to assume a functional relationship between E[L] and the expected loss l because the later data is usually available. It is well known that for catastrophe risk, there is a converse relationship between the conditional expected loss and catastrophe occurrence probability. The relationship is usually called power law in natural disaster science (see, e.g., Newman, 2005 and Embrechts et al, 1999) and for convenience we will rewrite the law instead as a relationship between the conditional expected loss and l as E[ L] 1l 2 with 1 , 2 being two positive constants given by the nature of the catastrophe. With this relationship, the approximation spread premium ratio (3.8) then becomes: 12
~ l 1 [ 1l 2 )]1/ 1 b0l b1 , l
(3.9)
with b0 (1 )1/ , b1 2 / . The above formula (3.9) can also be rewritten as ~ l l (1 b0l b1 ) .
(3.10)
Our proposed model thus complements the traditional robust control model with a new specification of the penalty function to control the ambiguity aversion magnitude and then derive some new catastrophe linked securities pricing formulas. In the next section, we present empirical tests to determine the adequacy of our specific CAT bonds pricing formula (3.10), which hereafter is also referred as Robust Control Formula, and will use these tests to compare our model with some other popular CAT bonds pricing approaches proposed in the previous literature. 4. Empirical test of CAT bonds pricing We now investigate the adequacy of the above specified CAT bonds pricing formula (3.10) by empirical test. Although CAT bonds are not publicly traded, there is an active nonpublic secondary market that provides some guidance on yields. The empirical data used in our test is collected from the quarterly and annual reports published by Lane Financial LLC between 1997 and 2016, which provides CAT bonds information such as spread premium and expected loss as well as sponsor, SPV, rating grades, issue date and maturity, probability of first loss, principal amount, with the covered territory, peril and trigger mechanism sporadically available. Regarding covered territories, about half of the CAT bonds in the samples cover the United States, about one quarter are multi-territory, and the rest cover other areas. In terms of underlying peril, about half are multi-peril CAT bonds, and the rest are mainly wind-specific or earthquake-specific CAT bonds (see, e.g., Trottier et al, 2016). 60
Spread Premium(%)
50 40 30 20 10 0 0
5
10
15
20
Expected Loss(%)
Figure 1 Illustration of the empirical data set, which consists of 360 CAT bonds issued between 1997 and March 2016. Source: Lane Financial. Our data set has removed several potential outliers and totally consists of 360 CAT bond tranches, and is illustrated in Figure 1 as a scatter plot of the spread versus 13
the expected looss, with su ummary stattistics as follows: the average a spre read premiu um is %, and the raatio of the sspread prem mium 6.777%, the aveerage expectted loss ratiio is 2.03% to thhe expectedd loss ratio of bond priincipal rang ges from 1.2 27 to 175, w with the aveerage of 66.72 and thhe mean of 4.12. It shhows that CAT C bonds tend to be issued for low probbability eveents, rangin ng from 1 to 3% exp pected loss. This refleects the higher capaacity of reiinsurance for f smaller, more freq quent eventss and the eeffectivenesss of cataastrophe riskk securitizattion especiaally for low frequency disasters. The secondary markeet yields onn CAT bond ds are show wn quarterlyy from the third t quarrter of 20001 through the first qquarter of 2016 2 in Fig gure 2. Figg. 2 showss the expected loss, the spread premium oover LIBOR R, and the bond b spreadd ratio (i.e., the ratioo of spreadd premium to expectedd loss), bassed on averrages of seccondary maarket trannsactions. Cummins and Barrieu (22013) pointted out from m a similar ffigure that CAT C bonnd spreads have h declineed significanntly as the market m has matured. F or examplee, the ratioo of premiium to exp pected loss was aroun nd 6 in 200 01. But thee spreads have h decllined steadiily until 20 005 as the market tigh htened follo owing Hurrricanes Katrina, Ritaa, and Wilm ma, with thee ratio of sppread premiium to expeected loss inncreasing to o 3.8 in thhe second quarter q of 2006. 2 Howeever, spread ds returned to lower leevels, fallin ng to abouut 2.2 in thhe fourth qu uarter of 20007 and firsst quarter of o 2008. Sppreads spikeed in respponse to thee 2008 finaancial crisiss, and rose to about 6 by the seccond quarteer of 20009. As the market m reco overed from m the crisis, spreads have again ffallen and have h beenn in the range of 2.0–3.0 from 20113 to the first quarter of 2016.
Figuure 2 Expeected lossess, spread prremiums, and a the prem mium/expeccted loss raatios baseed on averaages of secondary markket transactiions from 2001 throughh 2016. Sou urce: Lanne Financiall. Since Figuure 2 is baseed on the avverage transaction resu ults, it does not disting guish wheether the chhange of Preemium/Exppected Loss ratios overr the years ccomes from m the channge of propportions of catastrophee bond issu ues for diffferent expeccted losses. We thuss further illuustrate the fluctuationss of CAT bond prices over time iin Fig. 3, which w plotts the ratioss of the sprread premiuum to the ex xpected losss with resppect to diffeerent 14
expected losses for 2009–2016. It appears that premium spread is more pronounced for CAT securities with lower expected losses (i.e., lower probability that a contingent loss payment to the security issuers will be triggered): The ratios of the spread premium to expected loss are clearly higher for lower frequency, higher severity events, for example, the ratio ranges from 2.5 to 14 for the expected loss of 1% but ranges only from 2 to 4 for a loss on line of 7%. The ratios of the spread premium to expected loss also fluctuate dramatically depending upon the cycle phase, e.g., for the expected loss of 2%, the ratio of the spread premium to expected loss went from about 7 during the hard market of 2009-2011 to about 2.5 during the soft market of 2013-2016. An interesting fact is that if we augment the data set of the CAT bonds that were issued between June 1997 and 2001 into the Figure 3, contrary to the assertion of Cummins and Barrieu (2013) that CAT bond spreads have declined as the market has matured, it appears that the ratios of the spread premium to the expected loss during 1997-2001 are at about the same level as those in periods of soft markets during 2013-2016. In the earliest days of the market CAT bonds had been notorious for having high spreads. However, it seems from the Fig. 3, spreads in 1997-2001 are not especially large compared with those in later times and the ostensible higher average of the spread/expected loss ratios of about 9.03 (while the average ratio of the total 360 CAT bond is about 6.7) may be just due to the fact that more proportion of low frequency CAT bonds are issued during 1997-2001. 2009
2010
2011
2012
2013
2014
2015
2016
1997‐2001
20
18
16
Spread Premium/Expected Loss
14
12
10
8
6
4
2
0 0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
14.00%
16.00%
Expected Loss
Figure 3 Ratios of premium spreads to expected loss against expected loss during 2009-2016 and 1997-2001. Source: Lane Financial. Following Braun (2015) and Trottier et al (2016), we now use the above data to do the empirical tests of our model and also consider several alternative CAT bonds pricing models proposed in the previous literature to compare to our formula. The first and most simple alternative assumes the spread is a linear function of the expected loss: 15
~ l b0 b1l .
(4.1)
This specification assumes that the spread consists of a fixed b0 plus a fixed multiple b1 of the expected loss. The so-called Expected Value Premium Principle of ~ the form l (1 )l , which is commonly used in actuarial science, can be looked as a special case of the above linear function formula thus formula (4.1) is easy to understand and explain to bondholders. The second alternative we consider is a quadratic extension and is introduced to account for the presence of nonlinearity form as a polynomial: ~ (4.2) l b0 b1l b2l 2 . The third alternative below has been investigated in Braun (2015) and Trotter et al (2016), and was originally developed by the ILS fund Fermat Capital: ~ (4.3) l l b0 l (1 l ) , where the parameter b0 can be interpreted as a kind of “ILS Sharpe ratio”. The fourth alternative assumes the spread to be a power function of the expected loss and was originally introduced by Major and Kreps (2002): ~ (4.4) l b0l b1 . All the above formulas rely exclusively on pure actuarial or statistical models. The last alternative comes from Trottier et al (2016) and is derived from a utility-based economic model under hyperbolic absolute risk aversion (HARA) assumption and is called HARA formula: ~ (4.5) l l (1 b0 ln(l )) b1 . One possible problem with the HARA formula is that when the expected loss is very small, it may not be properly defined thus cannot be used to price the very low frequency insurance risk. We now investigate the empirical properties of various CAT bonds pricing formulas, and compare them with our robust control model based formula (3.10). We first calibrate all the pricing formulas by means of ordinary least-squares method, using the data set of CAT bonds that were issued between June 1997 and 2001, in which year September 11 attack happened. The estimated coefficients are presented in Table 1 for each pricing formula. The mean squared error (MSE) and the adjusted R2 of each regression are also displayed. We see that the highest in-sample accuracy is achieved by the quadratic formula, which has the smallest MSE and the largest adjusted R2. The performance of our model is only better than the Fermat Capital and Major-Kreps formulas. Figure 4 is a graphical illustration of the fitted regression curves of the aforementioned five approaches together with our robust control model based specification. Table 1: Regression results for our formula (3.10) and some alternative models based 16
on tthe empiricaal data of bo onds issuedd between 19 997 and 200 01. Linear
~ l b0 b1l
tic Quadrat ~ l b0 b1l b2 l 2
Ferrmat Capital ~ l l b0 l (1 l )
Major-Kreps ~ l b0 l b1
HARA ~ l l (1 b0 ln(l ))) b
Robust Co ontrol ~ l l (1 b0 l b1 )
b0
3.14
3.63
0.443
5.27
0.18
4.11
b1
1.98
0.85
0.33
0.38
0.81
b2
1
0.32
MSE 2
Adj. R
2.17
2.12
3.009
2.96
2.18
2.46
0.60
0.61
0.443
0.45
0.59
0.55
A reliable pricing mo odel needs tto build upo on strong ecconomic rellationships such thatt its overaall fitting behavior b reemains rob bust acrosss different data samp ples. Folllowing Braaun (2015)) and Troottier et all (2016), we thus iinvestigate the out--of-sample performancce of the six specific pricing formulas. f B Because off the cycllical behaviior of CAT bond pricees, we will only o use thee empirical data set off soft marrket betweenn 2014 and March 20116, which co onsists of 101 CAT boond tranches for our cross-validdation test. (2) Quadraatic
(1)Linear
(3)Fermatt Capital
(4)Maajor-Kreps
(5) HARA A
（6）Robu ust Control
17
Figure 4: Illustration of the fitted models using the data set of CAT bonds issued between June 1997 and 2001 based on the ordinary least-squares method. Our robustness test applies the parameters of each model in Table 1 estimated from the data between June 1997 and 2001 to cross-validate the out-of-sample accuracy of the model using the data set between 2014 and March 2016. The following commonly used out-of-sample performance measures are used: the mean absolute error (MAE), the root mean squared error (RMSE) and the out-of-sample R2. The results of our cross-validation test are presented in Table 2 and it shows that the best out-of-sample performance is achieved by our model. The superiority of our model’s performance in the cross-validation robustness test may be just due to the fact that our formula is based on a robust control theory. Another interesting fact is that the robustness performance of Major-Kreps formula purely based on statistical model is almost as good as our Robust Control approach. It may be because that although the Major-Kreps pricing formula just comes from practice in insurance industry, it has intuitively taken consideration of the suspect about the model misspecification and has formalized the taste for robustness into decision-making. Just as Hansen and Sargent (2016) mentioned recently, a robust planner's worst-case model responds to the forms of model ambiguity partly by having more persistence than the baseline or reference models, the robust control approach acquires tractability because the worst-case model turns out to be a time-invariant model in which projections for long-term behavior are more cautious and stochastic behavior is more persistent than in the baseline model. Table 2: Out-of-sample pricing performance in the cross-validation for our model and some other alternative models: All models parameters are estimated on the bonds issued during 1997-2001 and then applied to 2014-March 2016 empirical data. The tabulated numbers are mean absolute errors (MAE), root mean square errors (RMSE) and the out-of-sample R-squared.
Linear
Quadratic
Fermat Capital
Major-Kreps
HARA
Robust Control
MAE
2.84
4.52
3.27
1.42
3.59
1.49
RMSE
2.96
7.88
3.36
1.77
2.04
1.66
0.09
-5.42
-0.17
0.68
0.57
0.72
2
R
5. Conclusion Motivated by the fact that agents are ambiguity averse to catastrophe risks, we introduce a new type of penalty function to propose a catastrophe pricing model in a robust control framework. The cross-validation test shows that our model seems superior to the other alternative models with respect to model robustness tests. There are several extensions we may do to our current investigation. First, we have shown that the cyclical behavior of catastrophe risk market may be inferred from the time-varying model variation assumption but we have considered the model misspecification as a permanent psychological characteristic of the decision maker thus there is no learning in our model. On the other hand, an agent may learn 18
the model through successive approximations. It seems there is typically a price jump in catastrophe insurance market after a large CAT event occurs and Zhu (2011) assumed exogenously that the prior performance of CAT events might affect the magnitude of ambiguity aversion of decision makers to explain the empirical fact. It would be an important extension to incorporate forms of dynamic learning, for example, the robust learning theory, into our framework to give an endogenous explanation of the assumption and explore its implication to financial empirical evidence (see, e.g., Hansen, 2014 and Zhu et al, 2016). Second, we have used issue prices as the empirical data for fitting, which is at the risk of losing intra-year shifts in pricing. But using secondary market data instead to get a contemporaneous view will be at the risk that estimated loss data has shifted. It will be a challenging problem to get the updating of the expected loss as well as price data to test the dynamic behavior of our model. Thirdly, our equilibrium model exploits a representative paradigm. The representative agent is assumed to underwrite the total catastrophe risk and thus there is no insurability problem in our model, but an insurability problem may occur if agents in the economy are different. An introduction of heterogeneous agents might result in new insights about the role of robust control theory in catastrophe insurance research. It would be a very interesting topic to study the idiosyncratic degrees and weights of agents’ ambiguity aversion and its effect on insurance coverage window and the window shift after a large catastrophe event occurs. Finally, the robust control theory of Hansen and Sargent (2010) has shown how the model and state estimation misspecification emerges from two risk-sensitivity robust control operators. Since our CAT bonds pricing formula depends heavily on certain functional forms and parameters, it will be very interesting to study the possible relation of the two robust control operators with, e.g., the explicit form of the penalty function (2.6) we specified in Section 2 to examine the possible psychological meaning of the function form and for a better understanding of the superiority of our model’s robustness test performance.
19
References [1] Aase, K., 1999, An equilibrium model of catastrophe insurance futures and spreads, Geneva Papers on Risk and Insurance Theory，24: 69-96. [2] Arrow K. J., 1953, Le role des valeurs boursieres pour la repartition la meilleure des risques, in Econometrie, CNRS, Paris, 41–47. English version: The role of securities in the optimal allocation of risk-bearing, Review of Economic Studies, 1964, 31:91–96. [3] Braun, A., 2015, Pricing in the primary market for CAT bonds: New empirical evidence, Journal of Risk and Insurance. Article first published online: DOI: 10.1111/jori.12067. [4] Borch K., 1962, Equilibrium in a reinsurance market, Econometrica 30:424–444. [5] Bühlmann, H., 1970, Mathematical Methods in Risk Theory, Springer, Berlin. [6] Bühlmann, H., 1980, An economic premium principle, ASTIN Bulletin, 11 (1): 13–22. [7] Cabantous, L., Hilton, D., Kunreuther, H. and Michel-Kerjan, E., 2011, Is imprecise knowledge better than conflicting expertise? Evidence from insurer’s decisions in the United States, Journal of Risk and Uncertainty, 42: 211-232. [8] Cairns, A. J., 2000, A discussion of parameter and model uncertainty in insurance, Insurance: Mathematics and Economics, 27: 313–330. [9] Chen Z. and Epstein L., 2002, Ambiguity, risk, and asset returns in continuous time, Econometrica, 70(4): 1403-1443. [10] Cummins J. D. and Pauline Barrieu, 2013, Innovations in insurance markets: hybrid and securitized risk-transfer solutions, In: Dionne, G. (Ed.), Handbook of Insurance, Published by Springer, 547-603. [11] Embrechts P., Kluppelberg C. and Mikosch T., 1999, Modelling Extremal Events, Published by Springer. [12] Epstein, Larry G., and Martin Schneider., 2003, Recursive multiple-priors, Journal of Economic Theory, 113 (1): 1–31. [13] Froot, K., 1999. Introduction. In: Froot, K. (Ed.), The Financing of Catastrophe Risk, Published by University of Chicago Press, Chicago, 1–22. [14] Gerber, H.U., Shiu, E.S.W., 1994, Option pricing by Esscher transforms, Transactions of the Society of Actuaries XLVI, 99–191. [15] Hansen, L.P., 2014, Nobel Lecture: uncertainty outside and inside economic models, Journal of Political Economy, 122 (5): 945-987. [16] Hansen, L.P. and Sargent, T.J., 2001, Robust control and model uncertainty, American Economic Review, 91 (2): 60–66. [17] Hansen, L. P. and Sargent, T. J., 2010, Fragile beliefs and the price of uncertainty. Quantitative Economics, 1: 129–162. [18] Hansen, L.P. and Sargent, T., 2016, Sets of models and prices of uncertainty, working paper available at: http://www.nber.org/papers/w22000. [19] Harrington, S.E., Niehaus, G. and Yu, T., 2013, Insurance price volatility and underwriting cycles, In: Dionne, G. (Ed.), Handbook of Insurance, Published by Springer, 647-666. [20] Ju, N. and Miao J., 2012, Ambiguity, learning, and asset returns, Econometrica, 80: 559-591. [21] Keynes, J.M., 1921, A Treatise on Probability, Published by Macmillan. [22] Klibanoff, P., Marinacci, M., & Mukerji, S., 2005, A smooth model of decision making under ambiguity. Econometrica, 73(6): 1849-1892. [23] Knight, F.H., Risk, Uncertainty and Profit, 1921, Published by Houghton Mifflin. 20
[24] Lane, M. N., 2016, Trace data twenty one months on - ILS trade or quote data, Lane Financial, available at http://www.lanefinancialllc.com/content/blogcategory/910/84/. [25] Liu J., J. Pan and T. Wang, 2005, An equilibrium model of rare-event premia, Review of Financial Studies 18(1): 131-164. [26] Major, J. A. and Kreps, R. E., 2002, Catastrophe risk pricing in the traditional market, In Lane, M. N., (Ed.), Alternative Risk Strategies, chapter 10, Published by Risk Books. [27] Newman, M.E.J., 2005, Power laws, Pareto distribution and Zipf’s law, Contemporary Physics, 46 (5): 323–351. [28] Swiss Re, 2016, ILS market update, available at http://www.swissre.com/library/. [29] Trottier D-A, Lai V. S., and Charest A. C., 2016, CAT bond spreads: A model with HARA-utility and an empirical validation using nonparametric tests, working paper，Laval University, Canada, available at SSRN: http://ssrn.com/abstract=2781719. [30] Wang, S. S., 2004, CAT bond pricing using probability transform, Geneva Papers: Etudes et Dossiers, special issue on insurance and the State of the Art in CAT Bond Pricing, 278: 19-29. [31] Yang, H., 2006, Esscher Transform, In: Teugels J. L. and Sundt B. (Ed.), Encylopedia of Actuarial Science, Published by John Wiley and Sons. [32] Zhu, W., 2011, Ambiguity aversion and an intertemporal equilibrium model of catastrophe-linked securities pricing, Insurance: Mathematics and Economics, 49: 38-46. [33] Zhu, W., H. Kunreuther and E. Michel-Kerjan, 2016, Ambiguity types, robust learning and natural
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insurance:
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long-term
10.2139/ssrn.2113828.
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contracts
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help ， DOI:
PII: DOI: Reference:
S0167-6687(17)30045-8 http://dx.doi.org/10.1016/j.insmatheco.2017.08.006 INSUMA 2378
To appear in:
Insurance: Mathematics and Economics
Received date : January 2017 Revised date : August 2017 Accepted date : 19 August 2017 Please cite this article as: Zhu W., Wanting robustness in insurance: A model of catastrophe risk pricing and its empirical test. Insurance: Mathematics and Economics (2017), http://dx.doi.org/10.1016/j.insmatheco.2017.08.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Wanting robustness in insurance: A model of catastrophe risk pricing and its empirical test Wenge ZHU* School of Finance Shanghai University of Finance and Economics Shanghai, 200433, P. R. China
Abstract: Motivated by the fact that a lack of information about natural disasters may lead agents to be ambiguity averse to catastrophe risks, we introduce a new type of penalty function and propose an adjusted equilibrium model based on the function by allowing agents to act in a robust control framework against model misspecification with respect to rare events. The pricing formulas are then derived for various catastrophe linked securities such as catastrophe futures, options and bonds. We also estimate and test the model using empirical data of catastrophe bonds and compare it with various other models and investigate the robustness performance of alternative pricing formulas. Key Words: Ambiguity Aversion; Robust Control Theory; Catastrophe Risk Pricing JEL Classification: G12 G13 G28
* Tel: +86-21-65908401; e-mail: [email protected]
1. Introduction Economic losses from natural catastrophes such as hurricanes, earthquakes, and floods have increased significantly in China as well as in other parts of the world in recent years. There are two main factors driving the increasing losses from natural disasters: degree of urbanization and value at risk. These have not only generated greater insurance exposure levels but also rendered past economic and insured losses less relevant when attempting to estimate future scenarios. Usually losses from natural disasters are financed using one of five mechanisms: private insurance, capital market securities, government assistance, self-insurance, and charity. Although private insurance system is widely regarded as a more efficient mechanism for financing and managing the pure risk, it seems that insurance coverage is extremely limited for catastrophe losses and many exposures faced by corporate and household sectors never even reach insurers. For example, the 2008 Wenchuan Earthquake in China resulted in an estimated RMB 1.6 billion in insured claims, or only 0.2% of the total losses caused by the earthquake, while property insurers even in the United States have only provided an estimated $57 billion or 32% of total losses for the Hurricanes Katrina, Rita and Wilma in 2005. The paucity of insurance funding for large event risk has been considered as a barrier to better sharing of catastrophe risk and there are a lot of discussions among practitioners, academics, and policymakers on why there is a paucity of catastrophe risk sharing by the insurance sector, what are the weaknesses of the current system of financing disaster risk, and what kinds of alternative methods may be created to help solve the problem of inadequate catastrophe risk sharing (see., e.g., Froot, 1999 and Cummins and Barrieu, 2013, for reviews of such discussions). One of the most significant innovative catastrophe risk financing techniques has been through new financial instruments like CAT futures, options and bonds that bring catastrophe exposures directly to the capital market. One major driver of nonlife securitization is the so-called insurance underwriting cycle, which means that catastrophe insurance and reinsurance market usually undergoes alternating periods of soft markets, when prices are relatively low and coverage is readily available, and hard markets, when prices are high and coverage supply is restricted (Harrington et al, 2013). The previous literature has provided some explanations for the observed cycle patterns in the insurance market. For example, Froot (1999) examines eight potential explanations and concludes that the most compelling explanation is that holding capital is costly for insurers and reinsurers because of informational asymmetries between firms and the capital market. It is thus expected that the catastrophe securitization which provides risk exposure direct access to capital market has the potential to moderate the effects of the insurance underwriting cycle. Among all catastrophe financial products, it seems insurance-linked CAT bonds are by far the most successful securitized risk hedging instrument. Nevertheless, the CAT bonds market is still not immune to the underwriting cycle and the cyclical behavior of the CAT bonds market is very similar to a “typical” reinsurance market during the alternating periods of soft markets and hard markets. The cyclical behavior 1
of CAT bond prices seem to be puzzling in view of the fact that capital markets are not exposed to the imperfections of insurance markets. Zhu (2011) pointed out two other stylized facts of CAT bond prices that seem contradictory to traditional capital market theories. One is that the spread premium of “zero beta” CAT-linked bonds seems very high relative to the expected principal loss, the other is that spread premium is more pronounced for CAT securities with lower expected loss ratios thus generate a kind of “smirk” pattern in the cross-sectional plot of the premium to E[loss] ratio against the expected loss ratios. It is thus interesting to explain the facts by extensive theoretical economic analysis. The academic research on the role of insurance sector usually starts by examining a perfect economic model initially developed by Arrow (1953), with a specific extension to insurance market by Borch (1962) and Bühlmann (1980). The efficiency of the Arrow-Borch model for insurance risk financing depends on the following conditions: 1. There are a large number of agents in the economy and agents are expected utility maximizers, each with an increasing and concave utility function. 2. There is no transaction cost. 3. Either all agents face similar risks, or differences in risk can be observed at no cost and be incorporated in insurance premiums. 4. Insurance does not dissuade those who are insured from avoiding risks. The prevention efforts of agents in the economy are observable at no cost. 5. The corporate form for insurers is efficient. 6. The insured losses can be modeled with precision. In many insurance markets, one or several of the six conditions may not hold. Violations of the first and sixth condition are particular features of catastrophe risk market. While model misspecification is already a big problem for non-catastrophe risks which are extensively modeled, the catastrophe risk modeling is greatly hamstrung by the paucity of quality data, and this is further aggravated by lack of catastrophe models for different types of natural disasters in different parts of the world. Limitations on data and assumptions about a catastrophe model affect the model estimates and thus there is significant uncertainty or ambiguity associated with these estimates. Controlled experiments with underwriters and actuaries reveal that insurers would violate the axioms of expected utility theory and charge more premiums, in other words, insurers appear to be ambiguity averse if there is considerable uncertainty with respect to risk (Cabantous et al, 2011). Ambiguity aversion is not a new concept in economic theory and may be traced back to Knight (1921) and Keynes (1921). The basic point is that while a probability measure can represent likelihood assessment, it cannot also model the other dimensions of beliefs such as confidence of the likelihood assessment. Since it is impossible to distinguish the so-called ‘Knightian uncertainty’ or ‘ambiguity’ from ‘risk’ within the traditional subjective expected utility framework, it becomes important to formulate a new theory that permits a distinction between risk aversion and ambiguity aversion. In recent years, there have been many researches in financial economics literature 2
recognizing the presence of ambiguity and illustrating its potential role in explaining several asset market empirical puzzles (see, e.g., Chen and Epstein (2002), Klibanoff et al (2005), Liu et al (2005) and Ju and Miao (2012)). Ambiguity aversion is also not new to insurance and actuarial professionals. In fact, it may be categorized into the credibility theory in actuarial science which provides a framework examining the effect of learning the imprecise knowledge about insurance fundamentals by successive approximations (Bühlmann 1970). See, for example, Cairns (2000) for an exposition of the problem concerning both parameter and model uncertainty in the traditional actuarial learning approach. But just as Savage’s decision theory is inconsistent with the Ellsberg paradox evidence of ambiguity aversion, the traditional actuarial learning approach also raises two kinds of concern about model misspecification: one is that the chosen models set might be wrong or that some other fitting models might be better; the other is that even when we can specify the set of alternative models, we might have doubt how to choose and estimate a model from the set to fit the data. Zhu (2011) adopted the so-called robust control framework introduced in Hansen and Sargent (2001) and takes the choice patterns in the Ellsberg paradox or ambiguity aversion into consideration to explain the above-mentioned three stylized facts concerning catastrophe linked securities pricing. In the robust control settings, the perspective of a decision maker differs substantially from that in the traditional learning theory. The agent is assumed to deal with model uncertainty as follows: First, having noticed the unreliable aspects of the reference model based on existing information, he evaluates the future prospects under alternative models. Second, acknowledging the fact that the reference model is indeed the best statistical characterization of the available information, he penalizes the choice of alternative model by a distance function measuring how far it deviates from the reference model. The robust control theory uses the concept of relative entropy as a distance function to define a kind of objective function of the robust decision-maker with the relative entropy as a penalty term measuring the ambiguity aversion magnitude of the agent. But Zhu (2011) finds that the ambiguity aversion premium emerging from the relative entropy penalty function choice in Hansen and Sargent (2001) seems too high for the purpose of CAT risk pricing, so he introduces a more general penalty function form which may be chosen to fit empirical test. However Zhu was unable to give an explicit form of the function. So in this paper we will introduce a new concrete explicit form of the penalty function and propose an adjusted equilibrium model based on the function by allowing agents to act in a robust control framework against model misspecification with respect to rare events. The rest of the paper is organized as follows. Section 2 sets up the equilibrium model by introducing the new penalty function in the robust control framework. Section 3 demonstrates how to use the equivalent martingale measure derived in section 2 to value exchange-traded catastrophe securities and CAT bonds. Section 4 presents estimation and robustness test of the model using empirical CAT bonds data and gives a further discussion comparing our model with several previous related models. Section 5 concludes. 3
2. Robust Control Theory and Economic Model We first formulate our modeling framework of robust control theory and assume there is a catastrophe risk market where catastrophe loss Yt is traded and the market exchanges bundles of state-contingent contracts that can be analyzed as catastrophe-linked securities. The aggregate CAT losses Yt is assumed to follow a compound Poisson process; that is, dYt = Lt dN t ,
(2.1)
where Nt is a Poisson process with intensity λ>0 and Lt is the conditional random loss amount at time t, which is independent of Nt. For convenience we assume Lt is described by identical distribution function, Pr( Lt x ) FL ( x ) where FL(x) denotes the distribution function of a random variable L. A representative agent starts with an initial wealth w at time 0 and we further assume that the agent receive a premium of mc at that time to underwrite a part of CAT risk, where c is the price of the insurance covering total CAT losses between time 0 and a terminal, prespecified time T; m denotes the proportion the agent underwrites in the full insurance (we assume the proportion does not vary between time 0 and T for convenience here but the insurance proportion can also be allowed to adjust during the time interval). We assume all loss payments occur at time T and the continuously compounded risk-free interest rate r is a constant. Hansen and Sargent (2001) measure discrepancy between alternative model and reference model by ‘relative entropy’, defined as the expected value of a log-likelihood ratio (the log of the Radon-Nikodym derivative). More exactly, letting P=(Pt) be the probability measure associated with the reference model, the alternative ~ ~ ~ model be described by a probability measure P ( Pt ) with t dPt / dPt as the ~ Radon-Nikodym derivative of Pt with respect to Pt, the relative entropy is then
1~ E t [log( t )] , where the conditional expectation is evaluated 0 t
defined by I t lim
with respect to the density associated with the alternative model. Since it seems that the uncertainty aversion in practice only applies to the likelihood component of the CAT arrival, we effectively assume the agent in our model only has doubt about the probability of CAT events and the relative Radon-Nikodym derivative t is defined by the following stochastic differential equation:
dξt (e ht 1)ξt dNt (eht 1) λξt dt ,
(2.2)
where ht is a time dependent positive function controlling the model distortion magnitude, and 0 1 . By the construction, the process 4
{t , 0 t T } is a
~ ~ martingale of mean one. The measure P ( Pt ) thus defined is indeed a probability
measure, and we later denote P as the entire collection of such probability measures. Given the alternative model specification defined by (2.2), the distance measure It can be derived as: h h It= (e t ht e t 1) .
(2.3)
Using a penalty function of the relative entropy It, Zhu (2011) defines a kind of objective function of the robust decision-maker with the penalty function as a term measuring the ambiguity aversion of the agent. More exactly, in this generalized robust control framework, if we do not consider the randomness of consumption during the period, the agent’s time-t objective function is defined recursively by T
~ U(t)=U(W, t, y, m)= inf ~ {E t [u (WT ) (U ( s ), I s , s )ds ] Wt W , Yt y} , P P
(2.4)
t
with boundary condition U (W , T , y , m) u(W my ) and for convenience in the later paragraphs we further suppose u( x ) (1 / )e x , in which >0 is the risk aversion coefficient; WT
denotes the accumulated wealth of the agent at time T;
T
(U ( s), I , s)ds is s
a function of U and the distance measure Is that controls the
t
ambiguity aversion magnitude of the agent. More precisely, the penalty term in Zhu (2011) at time s is defined as:
(U ( s ), I s , s ) U ( s ) g s ( I s ) ,
(2.5)
where g s is assumed to be an increasing and convex function with g s (0) 0 so that T
T
t
t
(U ( s), I s , s)ds] = U ( s) g s ( I s )ds is indeed a penalty term. Zhu was unable to give an explicit form of the function but we find that if we define
g ( I s ) g ( (e hs hs ehs 1))
(ehs 1) 1 , ( 1)
(2.6)
in which both and are positive parameters, with associated with the general ambiguity aversion magnitude and the term associated with controls the distance function (eht ht eht 1) . It can be proved that the penalty form thus defined indeed 5
satisfies the increasing and convex function requirement and we later show it can exactly fit the empirical test for catastrophe risk pricing and discuss more about its possible economic meaning in the conclusion. With the above explicit form of the penalty term, the definition of the objective function can be formulated as a random optimization problem and satisfies the following HJB equation
Ut rWUW inf {e ht [EU (W , t , y L, m) U (W , t , y, m)] λγ ht
(e ht 1) β 1 U } =0, (2.7) α( β 1)
where Ut is the derivative of U with respect to t and UW is its first derivative with respect to W. With the above proper form assumption of the objective function, the first order condition for ht gives a constant solution given by ht h* ln(1 [ (
M L (m) 1
)]1 / ) ,
with ML denotes the moment generating function of the loss random variable L. (see, e.g., Zhu, 2011) Substituting the objective function into the HJB equation and the solution of Equation (2.7) at time 0 with W=w+mc, y=0 is given by
1
T
U ( w mc,0,0, m) exp{ ( w mc)e [e ( M L (m) 1) rT
h*
0
( e h* 1)
]dt} .
The first order condition for m (the second order condition also holds) in the above equation gives the following equation for the optimal insurance proportion m*: T
c e rT e h * E[ Lem* L ]dt e rT e h *E[ Lem* L ]T 0
(2.8)
Since in equilibrium the market is cleared and the representative agent will accept total catastrophe loss under the equilibrium price, in other words in equilibrium we should have m*=1, the solution to the market equilibrium and the corresponding EMM can then be summarized by the following proposition: Proposition 1: In equilibrium and under the assumptions above, the price of CAT risk is given by c e rT e h E[ LeL ]T ,
(2.9)
with
h ln(1 [ (
M L ( ) 1
)]1 / ) .
(2.10)
The pricing formula (2.9) can also be rewritten as an expectation with respect to an equivalent martingale measure (EMM) Q in the following equivalent form: c E Q0 [ e rT Y T ] E [ e rT Y T
exp( Y T hN T ) ] , E [exp( Y T hN T )]
(2.11)
with the Radon-Nikodym derivative of Q with respect to P given by dQ/dP 6
exp(YT hN T ) / E[exp(YT hN T )] .
Proof: Proof of Formula (2.9) is direct. See Appendix B of Zhu (2011) for the proof of the equivalent martingale measure expectation expression of (2.11). Now we apply the pricing kernel derived above to the pricing of more general catastrophe-linked securities. These kinds of securities may be looked as derivatives contingent on catastrophe losses. Let φ be a measurable function that specifies the payoff at maturity to the buyer of the catastrophe-linked securities, i.e., at time T the buyer receives φ(YT). By augmenting the new type of contingent claim contract in the above equilibrium model, it can be proved that the price of a contingent claim at time 0 with payoff φ(YT ) at time T is given by c E Q0 [ e rT (Y T )] E [ e rT (Y T )
exp( Y T hN T ) ]. E [exp( Y T hN T )]
(2.12)
Remark 1: We assume the insurance price is determined so that the agent underwrites the full insurance at time 0. The price may change in the later time to assure the agent’s total underwriting of the catastrophe risk. The difference of the EMM derived here from the traditional ones is that we have augmented an exponential of hNT to represent the ambiguity aversion of the agent (h depends on and according to Equation 2.10). Notice that h also depends on the distribution function of the random variable L, so if the distribution assumption of L varies in the future, we will induce a fluctuation of the spread premium coming from the ambiguity aversion thus the cyclical insurance price behavior may emerge endogenously from our model. Another way to make insurance price varies is to include learning into the robust control model, which Hansen and Sargent (2010) describes as robust learning theory. We will discuss more about it in the conclusion. Remark 2: If h becomes zero, i.e., if the market shows no ambiguity aversion, the form of dQ/dP then reduces to the traditional Esscher transform, which has been used for a long time in insurance and finance pricing (Bühlmann, 1980; Gerber and Shiu, 1994 and Yang, 2006). Wang (2004) shows that a generalized two-factor Wang Transform may be used for the pricing of CAT bonds by capturing economic behavior of greed and fear as well as risk aversion. It will be interesting to study whether the two-factor Wang Transform can be related with and grounded in our robust control economic framework. Remark 3: Epstein and Schneider (2003) criticized that robust control theory may cause dynamic consistency problem and propose a remedy to expand the alternative models set to attain what they called rectangular condition. But Hansen and Sargent (2016) pointed out that such an expansion may allow too much freedom in setting the alternative models for the use of a meaningful robust control problem. Since we have 7
assumed the conditional loss amount Lt as described by identical distribution function, the dynamic consistency problem may be solved by requiring ht as a constant in our alternative model specification defined by (2.2). 3. Catastrophe-linked Securities Pricing 3.1. Exchange-traded Catastrophe-linked Derivatives Pricing In this section we first apply the pricing kernel derived in the previous section to the pricing of exchange-traded catastrophe-linked securities. Hurricane Andrew in 1992 raised capital shock problem of the insurance and reinsurance industries and as a result, market participants began to explore alternative securitization measures for hedging catastrophic risk. The first such effort was the introduction of catastrophe futures and options by the CBOT in 1992. The 1992 contracts were replaced in 1995 with redesigned catastrophe call spread based on loss indexes provided by PCS, an independent insurance agency, but the CBOT-PCS contracts were delisted in 2000 due to lack of investor interest. There have been several recent efforts to relaunch securitization with payoffs triggered by catastrophic property losses. In 2007, futures and options contracts were introduced by the New York Mercantile Exchange (NYMEX), the Chicago Mercantile Exchange (CME), and the Insurance Futures Exchange (IFEX), whose contracts trade on the Chicago Climate Exchange (CCX). The IFEX futures are similar to the CBOT ones in using PCS index triggers but their contract payoff type is binary, while CME contracts are similar to the CBOT contracts in that they are not binary but are valued by a parametric index. Both types of contracts are subject to substantial basis risk and since it seems the problems with the failure of the CBOT contracts still have not been overcome with respect to the IFEX and CME products, it still remains unclear whether the new catastrophe derivatives market can succeed (Cummins and Barrieu, 2013). Since the contract structures are similar for the CBOT products with the new ones, we will only focus on CBOT contracts to illustrate the pricing of exchange-traded catastrophe-linked securities. Following Aase (1999), we make some simplifying assumptions that the catastrophe risk process Yt is a compound Poisson process with the distribution function of the claim size amounts L as identical Gamma distributed and then derive the explicit formulas for some specific forms of exchange-traded catastrophe -linked derivatives. Let the gamma distribution function be given by,
( x; a, b)
b a x a 1 by y e dy, x 0 . (a) 0
The moment generating function of the corresponding YT then becomes
M Y ( z, T ) E[exp(zYT )] exp[T [(
b a ) 1]] . bz
Hence the corresponding moment generating function with the EMM Q in Proposition 1 becomes 8
M Y ( z, T ; , h ) EQ [exp( zYT )] exp{Teh (
b a b a ) [( ) 1]} , b b z
(3.1)
which shows that the transformed process is of the same type, with parameters
( , a, b) replaced by (eh (
b a ) , a, b ) . b
Example 1: Catastrophe Futures Contract The first contract we consider is the catastrophe futures. We abstract from most details of the CBOT contracts and focus on the key mathematical features that enter into this type of contract. Define an insurance loss ratio index by the insured catastrophic property loss divided by , which is usually one-fourth the dollar value of the premium collected in the previous year. By CAT futures contract we mean a financial instrument with a stated value of $25,000 multiplied by the loss ratio index, i.e., the terminal payoff function φ(x) at time T can thus be described as
( x ) $25,000(
x F) ,
with F denotes the futures price determined at time 0. We then have the following futures pricing formula. Proposition 2: Under the assumption above, the following futures formula at time 0 is
F
eh ba 1aT . ( b ) a 1
(3.2)
Proof: From the above discussion about compound Poisson process with Gamma distributed claim size amount, YT under the equivalent martingale measure Q is also a compound Poisson process with Gamma distributed claim size. For the futures contract, we should have
0 E0Q [25,000(
YT F )] ,
by the formula (2.12), which implies that
F E0Q [
ehba 1aT 1 YT ] EQ0 [YT ] . ( b ) a 1
Example 2: Catastrophe Options Contract The second contract we consider is the catastrophe options. By this we mean a financial instrument with terminal payoff function φ(x) at time T described as
( x) $25,000(
x K )
with K denotes the strike price of the option. We then have the following pricing 9
formula for catastrophe options contract. Proposition 3: Under the assumption above, the following options pricing formula at time 0 is
(K ) e rT 25,000e
e h (
b a ) T b
( e h (
j 1
(
b a j ) ) b j!
ka [1 ( K , ja 1, (b )] K [1 ( K , ja, (b )]) (b )
(3.3)
Proof: Similar to the derivation in Proposition 2, the price of catastrophe options is given by
( K ) E0Q [e rT 25,000(
e rT 25,000e
e h (
YT K ) ]
b a ) T b
( e h (
j 1
where B(j) denotes the expected value of (
b a j ) ) b B( j ) , j!
L1 L2 ... L j
K ) under the adjusted
Gamma distribution, which can be calculated as
B( j )
ja [1 ( K , ja 1, (b )] K [1 ( K , ja, (b )] , (b )
and by this we obtain the conclusion of the proposition. Example 3: Catastrophe Spreads Contract In 1995 CBOT catastrophe futures and options contracts were replaced by catastrophe call spreads based on loss indexes provided by PCS. Again we abstract from the CBOT-PCS contract details and focus on its mathematical features. For this contract, the terminal payoff function can be described as:
( x) $25,000[(
x x K1 ) ( K 2 ) ]
where K1 denotes the lower strike price and K2 denotes the upper strike price. By capping of the options the spreads contract will limit the risk of its investors. The market price of the contract will then be given by the following proposition. Proposition 4: Under the assumption above, the following spread price formula at time 0 is
( K1 ) ( K 2 ) ,
(3.4) 10
with ( K1 ) and ( K 2 ) as given by the formula in Proposition 3. Proof: It can be directly derived from the payoff function form of the spreads contract. 3.2 Catastrophe Bonds Pricing We now apply formula (2.11) to catastrophe bonds pricing and do not need Gamma distributed claim size assumptions any more. CAT bonds market started during the 1990s and has matured and become a steady source of capacity for insurers and reinsurers since then. The CAT bonds market grew from less than $1 billion per year in 1997 to $7 billion in 2007. Even though the market was affected by the subprime financial crisis of 2008, CAT bonds issuance in 2014 already exceeded 2007, which previously was the largest year on record (Swiss Re, 2016). CAT bonds are usually priced at spreads over LIBOR, meaning that investors receive floating interest plus a spread or premium over the floating rate. We assume for convenience of discussion that the form of CAT bonds is described as follows: The CAT bond is priced at p, which denotes the bond principal. If the loss for a single CAT event in the period (0, 1) is less than loss trigger B1, the agent will get ~ back his principal p, and spread premium plus interest p(er 1 l ) at maturity time ~ 1, in which r is the LIBOR and l denotes the spread premium rate. Once a CAT loss
exceeds the trigger, the agent will forfeit some or all the principal at time 1. The loss fraction f12(L) of the principal is in proportion to the conditional loss L in the range between the trigger B1 and a cap B2 and is given by f12 ( L ) Max[0, Min ( L B1 , B2 B1 )] /( B2 B1 ) . We assume the spread premium plus
the risk-free interest is guaranteed no matter whether any principal loss occurs. The expected value of the bond principal loss proportion is denoted as l. ~ The ratio of the spread premium l to the expected principal loss ratio l is then given by the following proposition. Proposition 5: Under above assumptions, the ratio of the spread premium to the expected principal loss proportion (i.e., the ratio of the modified expected principal loss to the expected loss) of the above CAT bond is given by h ~ l (1 e e M L ( ) ) E[ f12 ( L )eL ] , (3.5) l (1 e )E[ f12 ( L )]M L ( ) with h given by (2.10). Proof: The loss fraction f12 ( L ) of the principal is given by f12 ( L ) Max[0, Min ( L B1 , B2 B1 )] /( B2 B1 ) 11
(( L B1 ) ( L B2 ) ) /( B2 B1 ) .
(3.6)
Thus the cash flow the agent gets back at time 1 can be described as ~ (Y1 ) p (er l I ( N1 0) f12 ( L)) , where I is an event indicator function, L denotes the first conditional catastrophe loss. The probability that at least one catastrophe loss occurs in the period is given by E[ I ( N1 0)] (1 e ) . The mean of the bond principal loss proportion is thus given by l E[ I ( N1 0) f12 ( L )] (1 e ) E[ f12 ( L)] . Now we apply the equivalent martingale measure Q in Proposition 1 instead to price the above CAT bond. The basic equation can be written as: ~ p EQ0 [ p e r ( e r l I ( N1 0) f12 ( L))] , from which we can get the formula to calculate the spread premium as below, ~ l E[exp(Y1 hN1 ) I ( N1 0) f12 ( L)] / E[exp(Y1 hN1 )] , which can be calculated as ~ (1 e e l
h
M L ( )
)E[ f12 ( L )eL ] , M L ( )
(3.7)
thus the spread premium to the mean principal loss ratio is as given in (3.5). From Equation (3.7), we can find that the spread premium emerging from risk aversion is relatively small and is not important for CAT bonds pricing (see, e.g., Zhu, 2011). So in later paragraphs, we will only focus on the effect of ambiguity aversion and assume the market is risk neutral to the catastrophe risk, in other words, we will assume 0 later. Under this assumption, the equation (3.5) then reduces to h ~ l (1 e e ) E[ f12 ( L )] 1 [E( L)]1 / . l (1 e )E[ f12 ( L )]
(3.8)
Given the lack of empirical data for E[L], we are unable to directly apply the formula (3.8) and thus are forced to assume a functional relationship between E[L] and the expected loss l because the later data is usually available. It is well known that for catastrophe risk, there is a converse relationship between the conditional expected loss and catastrophe occurrence probability. The relationship is usually called power law in natural disaster science (see, e.g., Newman, 2005 and Embrechts et al, 1999) and for convenience we will rewrite the law instead as a relationship between the conditional expected loss and l as E[ L] 1l 2 with 1 , 2 being two positive constants given by the nature of the catastrophe. With this relationship, the approximation spread premium ratio (3.8) then becomes: 12
~ l 1 [ 1l 2 )]1/ 1 b0l b1 , l
(3.9)
with b0 (1 )1/ , b1 2 / . The above formula (3.9) can also be rewritten as ~ l l (1 b0l b1 ) .
(3.10)
Our proposed model thus complements the traditional robust control model with a new specification of the penalty function to control the ambiguity aversion magnitude and then derive some new catastrophe linked securities pricing formulas. In the next section, we present empirical tests to determine the adequacy of our specific CAT bonds pricing formula (3.10), which hereafter is also referred as Robust Control Formula, and will use these tests to compare our model with some other popular CAT bonds pricing approaches proposed in the previous literature. 4. Empirical test of CAT bonds pricing We now investigate the adequacy of the above specified CAT bonds pricing formula (3.10) by empirical test. Although CAT bonds are not publicly traded, there is an active nonpublic secondary market that provides some guidance on yields. The empirical data used in our test is collected from the quarterly and annual reports published by Lane Financial LLC between 1997 and 2016, which provides CAT bonds information such as spread premium and expected loss as well as sponsor, SPV, rating grades, issue date and maturity, probability of first loss, principal amount, with the covered territory, peril and trigger mechanism sporadically available. Regarding covered territories, about half of the CAT bonds in the samples cover the United States, about one quarter are multi-territory, and the rest cover other areas. In terms of underlying peril, about half are multi-peril CAT bonds, and the rest are mainly wind-specific or earthquake-specific CAT bonds (see, e.g., Trottier et al, 2016). 60
Spread Premium(%)
50 40 30 20 10 0 0
5
10
15
20
Expected Loss(%)
Figure 1 Illustration of the empirical data set, which consists of 360 CAT bonds issued between 1997 and March 2016. Source: Lane Financial. Our data set has removed several potential outliers and totally consists of 360 CAT bond tranches, and is illustrated in Figure 1 as a scatter plot of the spread versus 13
the expected looss, with su ummary stattistics as follows: the average a spre read premiu um is %, and the raatio of the sspread prem mium 6.777%, the aveerage expectted loss ratiio is 2.03% to thhe expectedd loss ratio of bond priincipal rang ges from 1.2 27 to 175, w with the aveerage of 66.72 and thhe mean of 4.12. It shhows that CAT C bonds tend to be issued for low probbability eveents, rangin ng from 1 to 3% exp pected loss. This refleects the higher capaacity of reiinsurance for f smaller, more freq quent eventss and the eeffectivenesss of cataastrophe riskk securitizattion especiaally for low frequency disasters. The secondary markeet yields onn CAT bond ds are show wn quarterlyy from the third t quarrter of 20001 through the first qquarter of 2016 2 in Fig gure 2. Figg. 2 showss the expected loss, the spread premium oover LIBOR R, and the bond b spreadd ratio (i.e., the ratioo of spreadd premium to expectedd loss), bassed on averrages of seccondary maarket trannsactions. Cummins and Barrieu (22013) pointted out from m a similar ffigure that CAT C bonnd spreads have h declineed significanntly as the market m has matured. F or examplee, the ratioo of premiium to exp pected loss was aroun nd 6 in 200 01. But thee spreads have h decllined steadiily until 20 005 as the market tigh htened follo owing Hurrricanes Katrina, Ritaa, and Wilm ma, with thee ratio of sppread premiium to expeected loss inncreasing to o 3.8 in thhe second quarter q of 2006. 2 Howeever, spread ds returned to lower leevels, fallin ng to abouut 2.2 in thhe fourth qu uarter of 20007 and firsst quarter of o 2008. Sppreads spikeed in respponse to thee 2008 finaancial crisiss, and rose to about 6 by the seccond quarteer of 20009. As the market m reco overed from m the crisis, spreads have again ffallen and have h beenn in the range of 2.0–3.0 from 20113 to the first quarter of 2016.
Figuure 2 Expeected lossess, spread prremiums, and a the prem mium/expeccted loss raatios baseed on averaages of secondary markket transactiions from 2001 throughh 2016. Sou urce: Lanne Financiall. Since Figuure 2 is baseed on the avverage transaction resu ults, it does not disting guish wheether the chhange of Preemium/Exppected Loss ratios overr the years ccomes from m the channge of propportions of catastrophee bond issu ues for diffferent expeccted losses. We thuss further illuustrate the fluctuationss of CAT bond prices over time iin Fig. 3, which w plotts the ratioss of the sprread premiuum to the ex xpected losss with resppect to diffeerent 14
expected losses for 2009–2016. It appears that premium spread is more pronounced for CAT securities with lower expected losses (i.e., lower probability that a contingent loss payment to the security issuers will be triggered): The ratios of the spread premium to expected loss are clearly higher for lower frequency, higher severity events, for example, the ratio ranges from 2.5 to 14 for the expected loss of 1% but ranges only from 2 to 4 for a loss on line of 7%. The ratios of the spread premium to expected loss also fluctuate dramatically depending upon the cycle phase, e.g., for the expected loss of 2%, the ratio of the spread premium to expected loss went from about 7 during the hard market of 2009-2011 to about 2.5 during the soft market of 2013-2016. An interesting fact is that if we augment the data set of the CAT bonds that were issued between June 1997 and 2001 into the Figure 3, contrary to the assertion of Cummins and Barrieu (2013) that CAT bond spreads have declined as the market has matured, it appears that the ratios of the spread premium to the expected loss during 1997-2001 are at about the same level as those in periods of soft markets during 2013-2016. In the earliest days of the market CAT bonds had been notorious for having high spreads. However, it seems from the Fig. 3, spreads in 1997-2001 are not especially large compared with those in later times and the ostensible higher average of the spread/expected loss ratios of about 9.03 (while the average ratio of the total 360 CAT bond is about 6.7) may be just due to the fact that more proportion of low frequency CAT bonds are issued during 1997-2001. 2009
2010
2011
2012
2013
2014
2015
2016
1997‐2001
20
18
16
Spread Premium/Expected Loss
14
12
10
8
6
4
2
0 0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
14.00%
16.00%
Expected Loss
Figure 3 Ratios of premium spreads to expected loss against expected loss during 2009-2016 and 1997-2001. Source: Lane Financial. Following Braun (2015) and Trottier et al (2016), we now use the above data to do the empirical tests of our model and also consider several alternative CAT bonds pricing models proposed in the previous literature to compare to our formula. The first and most simple alternative assumes the spread is a linear function of the expected loss: 15
~ l b0 b1l .
(4.1)
This specification assumes that the spread consists of a fixed b0 plus a fixed multiple b1 of the expected loss. The so-called Expected Value Premium Principle of ~ the form l (1 )l , which is commonly used in actuarial science, can be looked as a special case of the above linear function formula thus formula (4.1) is easy to understand and explain to bondholders. The second alternative we consider is a quadratic extension and is introduced to account for the presence of nonlinearity form as a polynomial: ~ (4.2) l b0 b1l b2l 2 . The third alternative below has been investigated in Braun (2015) and Trotter et al (2016), and was originally developed by the ILS fund Fermat Capital: ~ (4.3) l l b0 l (1 l ) , where the parameter b0 can be interpreted as a kind of “ILS Sharpe ratio”. The fourth alternative assumes the spread to be a power function of the expected loss and was originally introduced by Major and Kreps (2002): ~ (4.4) l b0l b1 . All the above formulas rely exclusively on pure actuarial or statistical models. The last alternative comes from Trottier et al (2016) and is derived from a utility-based economic model under hyperbolic absolute risk aversion (HARA) assumption and is called HARA formula: ~ (4.5) l l (1 b0 ln(l )) b1 . One possible problem with the HARA formula is that when the expected loss is very small, it may not be properly defined thus cannot be used to price the very low frequency insurance risk. We now investigate the empirical properties of various CAT bonds pricing formulas, and compare them with our robust control model based formula (3.10). We first calibrate all the pricing formulas by means of ordinary least-squares method, using the data set of CAT bonds that were issued between June 1997 and 2001, in which year September 11 attack happened. The estimated coefficients are presented in Table 1 for each pricing formula. The mean squared error (MSE) and the adjusted R2 of each regression are also displayed. We see that the highest in-sample accuracy is achieved by the quadratic formula, which has the smallest MSE and the largest adjusted R2. The performance of our model is only better than the Fermat Capital and Major-Kreps formulas. Figure 4 is a graphical illustration of the fitted regression curves of the aforementioned five approaches together with our robust control model based specification. Table 1: Regression results for our formula (3.10) and some alternative models based 16
on tthe empiricaal data of bo onds issuedd between 19 997 and 200 01. Linear
~ l b0 b1l
tic Quadrat ~ l b0 b1l b2 l 2
Ferrmat Capital ~ l l b0 l (1 l )
Major-Kreps ~ l b0 l b1
HARA ~ l l (1 b0 ln(l ))) b
Robust Co ontrol ~ l l (1 b0 l b1 )
b0
3.14
3.63
0.443
5.27
0.18
4.11
b1
1.98
0.85
0.33
0.38
0.81
b2
1
0.32
MSE 2
Adj. R
2.17
2.12
3.009
2.96
2.18
2.46
0.60
0.61
0.443
0.45
0.59
0.55
A reliable pricing mo odel needs tto build upo on strong ecconomic rellationships such thatt its overaall fitting behavior b reemains rob bust acrosss different data samp ples. Folllowing Braaun (2015)) and Troottier et all (2016), we thus iinvestigate the out--of-sample performancce of the six specific pricing formulas. f B Because off the cycllical behaviior of CAT bond pricees, we will only o use thee empirical data set off soft marrket betweenn 2014 and March 20116, which co onsists of 101 CAT boond tranches for our cross-validdation test. (2) Quadraatic
(1)Linear
(3)Fermatt Capital
(4)Maajor-Kreps
(5) HARA A
（6）Robu ust Control
17
Figure 4: Illustration of the fitted models using the data set of CAT bonds issued between June 1997 and 2001 based on the ordinary least-squares method. Our robustness test applies the parameters of each model in Table 1 estimated from the data between June 1997 and 2001 to cross-validate the out-of-sample accuracy of the model using the data set between 2014 and March 2016. The following commonly used out-of-sample performance measures are used: the mean absolute error (MAE), the root mean squared error (RMSE) and the out-of-sample R2. The results of our cross-validation test are presented in Table 2 and it shows that the best out-of-sample performance is achieved by our model. The superiority of our model’s performance in the cross-validation robustness test may be just due to the fact that our formula is based on a robust control theory. Another interesting fact is that the robustness performance of Major-Kreps formula purely based on statistical model is almost as good as our Robust Control approach. It may be because that although the Major-Kreps pricing formula just comes from practice in insurance industry, it has intuitively taken consideration of the suspect about the model misspecification and has formalized the taste for robustness into decision-making. Just as Hansen and Sargent (2016) mentioned recently, a robust planner's worst-case model responds to the forms of model ambiguity partly by having more persistence than the baseline or reference models, the robust control approach acquires tractability because the worst-case model turns out to be a time-invariant model in which projections for long-term behavior are more cautious and stochastic behavior is more persistent than in the baseline model. Table 2: Out-of-sample pricing performance in the cross-validation for our model and some other alternative models: All models parameters are estimated on the bonds issued during 1997-2001 and then applied to 2014-March 2016 empirical data. The tabulated numbers are mean absolute errors (MAE), root mean square errors (RMSE) and the out-of-sample R-squared.
Linear
Quadratic
Fermat Capital
Major-Kreps
HARA
Robust Control
MAE
2.84
4.52
3.27
1.42
3.59
1.49
RMSE
2.96
7.88
3.36
1.77
2.04
1.66
0.09
-5.42
-0.17
0.68
0.57
0.72
2
R
5. Conclusion Motivated by the fact that agents are ambiguity averse to catastrophe risks, we introduce a new type of penalty function to propose a catastrophe pricing model in a robust control framework. The cross-validation test shows that our model seems superior to the other alternative models with respect to model robustness tests. There are several extensions we may do to our current investigation. First, we have shown that the cyclical behavior of catastrophe risk market may be inferred from the time-varying model variation assumption but we have considered the model misspecification as a permanent psychological characteristic of the decision maker thus there is no learning in our model. On the other hand, an agent may learn 18
the model through successive approximations. It seems there is typically a price jump in catastrophe insurance market after a large CAT event occurs and Zhu (2011) assumed exogenously that the prior performance of CAT events might affect the magnitude of ambiguity aversion of decision makers to explain the empirical fact. It would be an important extension to incorporate forms of dynamic learning, for example, the robust learning theory, into our framework to give an endogenous explanation of the assumption and explore its implication to financial empirical evidence (see, e.g., Hansen, 2014 and Zhu et al, 2016). Second, we have used issue prices as the empirical data for fitting, which is at the risk of losing intra-year shifts in pricing. But using secondary market data instead to get a contemporaneous view will be at the risk that estimated loss data has shifted. It will be a challenging problem to get the updating of the expected loss as well as price data to test the dynamic behavior of our model. Thirdly, our equilibrium model exploits a representative paradigm. The representative agent is assumed to underwrite the total catastrophe risk and thus there is no insurability problem in our model, but an insurability problem may occur if agents in the economy are different. An introduction of heterogeneous agents might result in new insights about the role of robust control theory in catastrophe insurance research. It would be a very interesting topic to study the idiosyncratic degrees and weights of agents’ ambiguity aversion and its effect on insurance coverage window and the window shift after a large catastrophe event occurs. Finally, the robust control theory of Hansen and Sargent (2010) has shown how the model and state estimation misspecification emerges from two risk-sensitivity robust control operators. Since our CAT bonds pricing formula depends heavily on certain functional forms and parameters, it will be very interesting to study the possible relation of the two robust control operators with, e.g., the explicit form of the penalty function (2.6) we specified in Section 2 to examine the possible psychological meaning of the function form and for a better understanding of the superiority of our model’s robustness test performance.
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