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2 Insurane: Mathematis and Eonomis 54 (2014) Contents lists available at SieneDiret Insurane: Mathematis and Eonomis journal homepage: Risk models with dependene between laim ourrenes and severities for Atlanti hurrianes Mathieu Boudreault a,, Hélène Cossette b, Étienne Mareau b a Université du Québe à Montréal, Département de mathématiques, C.P. 8888, su. Centre-Ville, Montréal, QC, H3C 3P8, Canada b Université Laval, Éole d atuariat, Québe, QC, G1V 0A6, Canada a r t i l e i n f o a b s t r a t Artile history: Reeived November 2010 Reeived in revised form September 2013 Aepted 5 November 2013 Keywords: Risk theory Hurriane risk Risk measures El Niño/Southern Osillation (ENSO) Florida hurrianes In the line of Cossette et al. (2003), we adapt and refine known Markovian-type risk models of Asmussen (1989) and Lu and Li (2005) to a hurriane risk ontext. These models are supported by the findings that El Niño/Southern Osillation (as well as other natural phenomena) influene both the number of hurrianes and their strength. Hurriane risk is thus broken into three omponents: frequeny, intensity and damage where the first two depend on the state of the Markov hain and intensity influenes the amount of damage to an individual building. The proposed models are estimated with Florida hurriane data and several risk measures are omputed over a fititious portfolio Elsevier B.V. All rights reserved. 1. Introdution Supported by natural and eonomi phenomena, there has been a rising interest in the risk theory literature on models linking laim ourrene and severity. Two important lasses of dependene between the latter two omponents are: (1) renewal models in whih the interarrival time and laim size are dependent (see for instane Albreher and Teugels (2006), Boudreault et al. (2006), Cossette et al. (2008), Badesu et al. (2009), Cheung et al. (2010) and referenes therein) and (2) risk models with a Markovian environment where interarrival times and laim sizes (and possibly premiums as well) all depend upon the state of a ommon Markov hain (see for example Asmussen (1989), Lu and Li (2005), Lu (2006) and Ng and Yang (2006) and referenes therein). For a general review of the latter models, see Chapter 7 in Asmussen and Albreher (2010). In the limatology and meteorology literature, it is well known that the phenomenon known as El Niño/Southern Osillation influenes both the number of hurrianes and their strengths (wind speed, amount of preipitation, et.) (see for example Gray (1984), Meyer et al. (1997) and Landsea and Pielke (1999); Pielke and Landsea (1998) among others). A dependene relationship between Corresponding author. Tel.: ; fax: addresses: boudreault.mathieu@uqam.a (M. Boudreault), helene.ossette@at.ulaval.a (H. Cossette), etienne.mareau@at.ulaval.a (É. Mareau). hurriane frequeny and intensity is thus obvious allowing Markovian models ited above to be adapted and refined to suit suh a hurriane ontext. Note that in all the aforementioned risk theory papers, the fous has been put mostly on deriving ruin measures. In this paper, we extend the previous lass of Markovian models in the line of Cossette et al. (2003) by deomposing natural atastrophe risk into frequeny, intensity (strength of the event) and damage. Based upon the literature in limatology and meteorology, we propose different joint hurriane frequeny and intensity models. We represent by a latent proess the urrent state of hurriane ativity, whih an be influened by many physial phenomena. We first introdue a Markov-swithing framework where both frequeny and intensity are allowed to be state-dependent. Models with two states or three states are onsidered. We also extend the frequeny models of Lu and Garrido (2006, 2005) suh that hurriane frequeny and intensity are dependent. Using Florida landfalling hurriane data (both frequeny and intensity) and ivil engineering approahes to quantify damage, we estimate and ompare the latter models in order to analyze various risk measures of a fititious portfolio of poliyholders. The paper is strutured as follows. Setion 2 introdues the general modeling framework for hurriane risk. In Setion 3 we detail the joint frequeny and intensity models proposed, while Setion 4 fouses on the damage omponent. In Setion 5, we apply the models to Florida data. Finally, Setion 6 ends the paper with a onlusion /$ see front matter 2013 Elsevier B.V. All rights reserved.

3 124 M. Boudreault et al. / Insurane: Mathematis and Eonomis 54 (2014) Table 1 Saffir Simpson hurriane intensity sale. Category Desription MSWS (m/s) Past ount 1 Minimal Moderate Extensive Extreme Catastrophi over Modeling hurriane risk 2.1. Introdution Aording to the National Hurriane Center (of the National Oeani and Atmospheri Administration (NOAA)) and to Federal Emergeny Management Ageny (FEMA), a hurriane is [... ] a type of tropial ylone, the generi term for a low pressure system that generally forms in the tropis. A typial ylone is aompanied by thunderstorms, and in the Northern Hemisphere, a ounterlokwise irulation of winds near the earth s surfae. Depending on the tropial ylone s loation or strength, a tropial ylone may be known as a hurriane, typhoon, tropial storm or depression (for more details, see Neumann (1993)). Hurriane ativity in the Atlanti Oean and on the Amerian East Coast is known to be influened by many phenomena suh as the Atlanti Multideadal Osillation (AMO) (see Chylek and Lesins (2008), and the NOAA Frequently Asked Questions), El Niño/Southern Osillation (ENSO) (see Gray (1984), Meyer et al. (1997) and Landsea and Pielke (1999); Pielke and Landsea (1998) among others) and limate hange (see Emanuel (2005) and WMO (2006)). For example, ENSO represents the ylial patterns observed in the surfae temperature of the Paifi Oean and its hanges in air surfae pressure. During yles that may last months or years, oean temperatures in the entral tropial Paifi Oean tend to warm (El Niño) and then ool (La Niña) in a ylial pattern. Aompanying these temperature variations are hanges in air surfae pressure aross the Paifi. During El Niño (La Niña), higher (lower) pressures are observed in the Western Paifi and lower (higher) pressures are observed in the Eastern part of the basin. The ylial hanges in air pressure is known as the El Niño Southern Osillation (ENSO). Many authors have reported that ENSO is known to have an important influene on hurriane frequeny and intensity in the Atlanti Oean (see aforementioned authors). Indeed, during La Niña, more hurrianes are generated (on average) and these hurrianes are generally stronger. This is the usually the opposite in El Niño. This means that frequeny and intensity are dependent omponents of hurriane risk. Note that frequeny represents the number of hurrianes that made landfall in a given region within a speifi time period (year, month). Intensity is defined as the strength of a hurriane at a given loation and is generally measured on the Saffir Simpson sale, whih is based upon wind speeds. The latter lassifies hurrianes aording to five levels (see Table 1 for a desription), whih are distinguished on the basis of the 1-min maximum sustained wind speed (MSWS). We mention that instruments measure the average wind speed within 1-min time intervals. The MSWS is the highest of these means. Moreover, the approah is not limited to the Saffir Simpson sale. One may also inlude less severe tropial storms or lassify hurrianes in more ategories based on the MSWS or other measures. When winds are between 18 and 32 m/s (meters per seond), the ylone is lassified as a tropial storm and below 17 m/s, the storm is a depression. In the latter ases, the Beaufort sale is used. In the fourth olumn of Table 1, we indiate the distribution of hurriane intensity, for hurrianes that made landfall in Florida. The dataset used to ompute the numbers in this olumn is disussed in Setion 5. Finally, damage is related to the amount of losses suffered by a poliyholder for a given hurriane. This omponent is losely linked to the intensity of the hurriane and is presented in more detail in Setion General modeling framework Let N = {N (t), t > 0} represent the ounting proess of the number of hurrianes that make landfall in a given region during the time interval (0, t]. Let also the random variable (r.v.) I k represent the intensity of the k-th hurriane, whih is, as disussed earlier, the strength of a given hurriane on a given sale. We also define the proess X i = {X i (t), t > 0} where X i (t) is the total amount of losses suffered by poliyholder i due to the N (t) hurrianes that ourred in (0, t] i.e. N(t) C X i (t) = i,k, N (t) > 0, (1) k=1 0, N (t) = 0. The amount of loss due to the k-th hurriane is defined by the r.v. C i,k with C i,k = U i,k b i, (2) where the salar b i is the exposure or the value of the insured building and the r.v. U i,k [0, 1] represents the proportion of damage. The information regarding the type of building and its onstrution will be embedded in U i,k. Moreover, the intensity of the k-th hurriane will influene the extent of damage to a property so that the onditional distribution of U i,k depends upon the intensity r.v. I k. The speifi relationship between U i,k and I k will be defined later in Setion 4. The way that we define C i,k assumes that losses to an individual building annot be larger than its given value b i, or in other words, C i,k [0, b i ]. The type of building and the fore of the hurriane will determine the distribution of C i,k and thus the total loss suffered by poliyholder i for hurriane k. For a portfolio of n poliyholders living in a hurriane-prone area, the proess for the aggregate losses is defined by S = {S (t), t > 0} where S (t) is the aggregate losses for the time period (0, t] e.g. n N(t) n C S (t) = X i (t) = i,k, N (t) > 0, (3) k=1 i=1 i=1 0, N (t) = 0. We interpret n i=1 C i,k as the aggregate amount of losses due to the kth hurriane. There are two soures of dependene within this model. First, the number of hurrianes and their intensities are ommon to all poliyholders of the same region. Seond, as mentioned in the Introdution, ENSO indues a dependene relation between hurriane frequeny and intensity. This is detailed next El Niño/Southern osillation As previously mentioned in Setion 2.1, various phenomena suh as AMO, ENSO and limate hange influene the hurriane ativity level. Although ENSO is observed via the Oeani Niño (ONI) and Southern Osillation Indies (SOI), the fat that many meteorologial phenomena interat to influene the hurriane ativity level (not just ENSO) justifies the use of a latent proess approah. However, to lighten the presentation, we will interpret the latent stohasti proess as ENSO with states orresponding roughly to El Niño and La Niña even though there might not be an exat orrespondene with the ONI and SOI. Thus, the interpretations that we attribute to the states of ENSO an be desribed

4 M. Boudreault et al. / Insurane: Mathematis and Eonomis 54 (2014) Fig. 1. Graphial summary of the various omponents of the model. as (low frequeny, low intensity)/(high frequeny, high intensity) states or El Niño/La Niña respetively. Let R = {R (t), t 0} be the proess that represents the timely evolution of ENSO and I = {I 1, I 2,...} be a sequene of r.v.s where I k is the intensity of the kth hurriane. In Setion 3, we define speifi models for R, N and I where R has a simultaneous influene on the latter two. Fig. 1 illustrates the various omponents of the model and their interations. Moreover, we suppose that given ENSO, frequeny and intensity are independent and time-independent. In other words, one we know the state of ENSO at time t, the intensity of hurrianes that ourred within a period is independent from the number of hurrianes that happened during the same period. Furthermore, one the evolution of ENSO is known, frequeny and intensity of hurrianes are serially-independent. Finally, we assume minimum ollateral damage meaning that if two neighbor buildings suffer damage, it is beause they were exposed to the same hurriane. This means that we exlude possibilities that may reate further dependene between buildings, apart from the ommon exposure to ENSO. Mathematially, this means that given ENSO, the number of hurrianes and the intensity I k of a given hurriane, the r.v.s U 1,k, U 2,k,..., U n,k are onditionally independent. 3. Joint frequeny and intensity models In this setion we onsider three models with a Markovian environment and a doubly periodi model to represent ENSO, along with models for the frequeny and intensity of hurrianes Three models with a Markovian environment Within the three models with a Markovian environment, R is assumed to be a latent disrete-time Markov hain where eah state defines the status of ENSO. Beause ENSO is a long-term phenomenon (duration of several years for example), transitions of R between states are assumed annual unless stated otherwise. Furthermore, in everything that follows, we assume that given the state of R, the onditional distribution of the intensity of the k-th hurriane is binomial. Hene, beause we use the Saffir Simpson sale on values {1, 2, 3, 4, 5}, we have that ( I k R (t)) 1 Binom 4, q R(t), (k = 1, 2,...), where the probability parameter q R(t) evolves with R. We expet q R(t) to be higher (lower) during La Niña (El Niño) so that severe hurrianes (4 and 5) will be more (less) likely. We present three different frequeny models: (1) a twostate Markov-swithing Poisson proess, (2) a three-state Markovswithing Poisson proess and (3) a two-state non-homogeneous Poisson proess Two-state Markov-swithing Poisson proess In the two-state Markov-swithing Poisson proess, we suppose that R is a latent two-state Markov hain suh that R (t) = 0 represents El Niño and R (t) = 2 denotes La Niña. Thus, hurriane frequeny is a Poisson proess suh that the rate of arrival of hurrianes at time t, denoted by λ R(t), depends on the value of R (t) Three-state Markov-swithing Poisson proess In the meteorology literature, a third state of oean temperatures has been onsidered in e.g. Landsea and Pielke (1999) and Katz (2002, 2008). This is known as a neutral state, that ours between transitions from La Niña to El Niño and vie-versa. Then, to take into aount this phenomenon, we propose in the seond model with a Markovian environment that R to be a latent Markov hain that follows a ylial pattern illustrated in Fig. 2. Note that one annot use a standard three state Markov hain beause of this type of yliality. Indeed, we annot observe an observation of El Niño followed by Neutral and then El Niño. One in a Neutral state, ENSO will eventually return to a La Niña phase. To aomplish this in a Markovian environment, we define a four state Markov model, with El Niño and La Niña states, and two Neutral states. However, the two Neutral states will have the same onditional frequeny and intensity. We denote by R (t) = 1o the neutral state of oean temperatures at time t given that at some time in the past, the transition to Neutral ame from El Niño. Similarly, we define R (t) = 1a to be the neutral state of surfae water temperatures at time t given at some time in the past, the

5 126 M. Boudreault et al. / Insurane: Mathematis and Eonomis 54 (2014) Fig. 2. Illustration of the evolution between the states in the three-state Markov-swithing proess. transition to Neutral ame from La Niña. Thus, the transition matrix P has 4 dimensions and beomes p 00 p 01o p P = 1o1o 0 p 1o2 p 1a0 0 p 1a1a 0. (4) 0 0 p 21a p 22 This transition matrix an be interpreted as follows. From El Niño, oean temperatures either remain in El Niño or swithes to a neutral phase. One in a neutral phase, R annot swith bak to El Niño and is fored to either stay in a neutral phase or move to La Niña. This means that a move from El Niño to La Niña or vie versa must be made using a transition to a neutral phase. This formulation of the ENSO yle is tehnially a 4-state Markov hain, but states 1o and 1a are both onsidered the neutral state. In this framework, we assume that hurriane frequeny is a Markovswithing Poisson proess with 3 states (sine λ 1o = λ 1a ) and intensity also relies on 3 different states Two-state non-homogeneous Poisson proess In Lu and Garrido (2006), R is also a two-state latent Markov hain. However, given the state of R, hurrianes arrive aording to a non-homogeneous Poisson proess. In their approah, R evolves on an annual basis, but the non-homogeneous Poisson proess aptures the fat that hurrianes our between June and November, whih is known as the hurriane season. They define an annual periodiity funtion λ (A) (t) (t [0, 1]) suh that 0, 0 t < 5 12, λ (A) (t) k T (t) αa 1 1 T (t) βa 1, 0, where 12 T (t) = t k 1 = T t αa 1 A 1 T t βa 1 A t = 5 A α A 1 12 α A + β A t 11 12, < t 1, The parameters α A and β A will determine the shape of λ (A) (t) i.e. when hurrianes are more likely to our within a year. Moreover, t A is the moment when λ(a) (t) reahes its mode, suh that λ (A) t A = 1. Beause the annual periodiity funtion λ (A) (t) varies within [0, 1], Lu and Garrido (2006) define the rate of arrival of hurrianes as λ (t) = λ (L) (t) λ (A) (t t ), t 0, where t is the floor funtion. The funtion λ (L) (t), defined as λ (L) λ (L) 1 (t) =, R (t) = 1 λ (L) 2, R (t) = 2, sales λ (A) (t) to get a representative rate of arrival funtion. We use the notation λ (A) and λ (L) to ontrast between the annual and long-term omponents of the rate of arrival of hurrianes. (5) 3.2. Doubly periodi model Lu and Garrido (2005) have proposed a doubly periodi nonhomogeneous Poisson proess for hurriane frequeny. We ombine this frequeny model with an intensity model to aount for dependene between these two omponents of hurriane risk. We first start by briefly summarizing their approah. Note that in the doubly periodi model, R is a deterministi proess. In the doubly periodi model, the short-term rate of arrival of hurrianes λ (A) (t) defined in (5) is affeted by R, whih is in fat a deterministi periodi funtion representing ENSO. The annual rate of arrival of hurrianes λ (A) (t) is multiplied by a onstant whih evolves periodially, aording to smooth transitions of R between El Niño and La Niña. Consequently, the intensity funtion λ (t) is suh that t λ (t) = λ t (L) + t A λ (A) (t t ), t > 0, where R (t) = λ (L) (t) is the long-term intensity funtion whih haraterizes ENSO, ( = 1, 2, 3,...) is the length of an ENSO yle, and λ (L) (t) is defined as λ (L) (t) = a + b a t ml t αl 1 ml k L t ml t βl 1 ml 1 for t > 0, where t L k L = m αl 1 L 1 t L m L and t L = m L + αl 1 α L + β L 2. βl 1 Moreover, t L is the mode of λ(l) (t), k L is a saling fator, a (b) represents the minimum (maximum) amplitude of peak values and m L represents the time when the lowest point is reahed by λ (L) (t) (desribed by Lu and Garrido (2005) as the starting point of the omplete long-term yle). The minimum a (maximum b) amplitude of peak values orresponds to El Niño (La Niña). Note that α L and β L determine the shape of the periodiity embedded in λ (L) (t), just like α A and β A do for λ (A) (t). Fig. 3 illustrates the path of the deterministi proess R defined by the doubly-periodi funtion, in the ontext of ounting the number of hurrianes over 10 years (from Lu and Garrido (2005)). The short-term urve provides the rate of arrival of hurrianes over eah year, i.e. λ (t) and emphasizes that they only our between June and November. The peak rate of arrival is shifted by a fator ( long-term urve or the funtion λ (L) ) that evolves slowly over the long-term, i.e. ENSO. Lu and Garrido (2005) solely fous on hurriane frequeny but one needs to relate the intensity with ENSO, whih is determined by λ (L) (t). To do so, we need to transform this [0, [ input into a valid parameter for the binomial distribution. This an easily be done using a transformation f : [0, [ [0, 1] suh as the umulative distribution funtion (.d.f.) of a positive r.v. Thus, our extension to aount for intensity is suh that I k follows a binomial

6 M. Boudreault et al. / Insurane: Mathematis and Eonomis 54 (2014) Table 2 Summary of the joint frequeny and intensity models proposed. Model ENSO Frequeny Intensity #1 2-state Markov hain #2 3-state Markov hain #3 2-state Markov hain #4 Deterministi proess M.-S. Poisson proess M.-S. Poisson proess M.-S. non-homo. Poisson proess Non-homo. Poisson proess M.-S. binomial distribution M.-S. binomial distribution M.-S. binomial distribution Binomial distribution q t = f λ (L) (t) Fig. 3. Illustration of a doubly periodi rate of arrival funtion. Soure: Taken diretly from Lu and Garrido (2005) s Figure 2. distribution over {1, 2, 3, 4, 5} with probability q t = f λ (L) (t). This will be illustrated in the numerial example of Setion 5. Finally, Lu and Garrido (2005) only represented hurrianes within a year and over yles that last approximately 3 5 years. One an also use the approah of Lu and Garrido (2005) to model AMO, whih an last deades, and ENSO, whih may last a few years Summary We provide in Table 2 a summary of all the joint frequeny and intensity models that we have proposed. Note that M.-S. stands for Markov-swithing. 4. Modeling hurriane damage 4.1. Introdution Aording to the FEMA, damage from hurrianes mainly omes from high winds and heavy rain but for buildings built along oastal areas, storm surges are severe threats. A storm surge is an abnormal rise of the sea level due to high winds, that will result in important floods. Damage to buildings an range from broken windows to ollapsing roofs or quasi-total destrution (sine the foundations will remain). For example, the National Oeani and Atmospheri Administration (NOAA) desribes the damage for frame homes resulting from a ategory 4 hurriane as: Poorly onstruted homes an sustain omplete ollapse of all walls as well as the loss of the roof struture. Well-built homes also an sustain severe damage with loss of most of the roof struture and/or some exterior walls. Extensive damage to roof overings, windows, and doors will our. Large amounts of windborne debris will be lofted into the air. Windborne debris damage will break most unproteted windows and penetrate some proteted windows. The damage model onsidered in this paper diretly links damage to wind speed and is based upon Unanwa et al. (2000a) and Unanwa and MDonald (2000b). Indiret damage that may ome from heavy rain into a damaged building (from the roof and broken windows) is also taken into aount. Thus, water infiltration from the ground is not aounted for, whih exludes damage from storm surges. The types of damage aounted for in these two papers are onsistent with homeowners insurane sine floods are not usually overed by insurane ompanies, but water that gets into the property from the roof and broken windows resulting from damage aused by high winds may be overed. Aording to the 2011 Florida Statutes, Title XXXVII, Chapter 627 and Setion 4025, Paragraph (2)(a): Hurriane overage is overage for loss or damage aused by the peril of windstorm during a hurriane. The term inludes ensuing damage to the interior of a building, or to property inside a building, aused by rain, snow, sleet, hail, sand, or dust if the diret fore of the windstorm first damages the building, ausing an opening through whih rain, snow, sleet, hail, sand, or dust enters and auses damage Wind damage bands The methodology developed in Unanwa et al. (2000a) relies on a list of building omponents that might fail beause of high winds. Those omponents are listed as: [... ] roof overing, roof struture, exterior doors and windows, exterior wall (inludes finishes, eletrial and mehanial omponents supported, ladding and support systems), interior (inlude ontents), 1 strutural system (inludes olumns, girders, elevated floors and onveying equipment) and foundation. Four different ategories of buildings have been analyzed in Unanwa et al. (2000a): 1 3 story residential, ommerial/ industrial, government/institutional and 4 10 story mid-rise buildings. Confidene (wind damage) bands as a funtion of wind speed for eah type of building are illustrated in Figs of Unanwa et al. (2000a). Using the Saffir Simpson sale (see Table 1), one dedues that for a 1 3 story residential building, the 95% onfidene interval for proportions of damage is [7%, 30%] when the winds are at 50 m/s, whih is a light ategory 3 hurriane Individual adjustments The wind damage bands were intended for a typial property within a very wide ategory. However, not all residential properties have been onstruted like the typial 1 3 story residential building desribed in Unanwa et al. (2000a). The approah presented in their seond paper, i.e. Unanwa and MDonald (2000b), allows for very exhaustive individual adjustments. As muh as 20 different riteria are introdued in this paper to differentiate residential buildings. Examples of riteria are roof overing, geometry and span, building ode and age, types of windows glass, et. Many experts have been gathered to evaluate the potential failure of eah building omponent to ompute the global relative 1 Interior ontents mean ounters, upboards, sinks, et., i.e. things that are inside the property and are physially attahed to the building. It does not inlude furnitures, applianes or eletronis.

7 128 M. Boudreault et al. / Insurane: Mathematis and Eonomis 54 (2014) resistivity index (RRI) of a building. This global index takes value between 0 and 1 and it is obtained by weighting the quality of eah omponent for a given building. Table 2 of Unanwa and MDonald (2000b) provides weights and quality fators for eah possible value of eah omponent. The weighted sum of quality fators for a partiular building provides its RRI. The value of RRI for a typial building is 0.5. If the value of RRI is lower (greater) than 0.5, it indiates that the building i is more (less) resistant than the typial building of its ategory. Assume that poliyholder i owns a property that belongs to one of the 4 building ategories (φ = 1, 2, 3, 4) with eah ategory having a given RRI i. Then aording to Unanwa and MDonald (2000b), Ui,k Ik = θ = l φ,θ + RRI i u φ,θ l φ,θ (6) whih is deterministi for θ = 1, 2,..., 5. Note that l φ,θ, u φ,θ is the 95% onfidene interval for proportions of damage for a building that belongs to ategory φ during a hurriane of intensity θ. The fat that U i,k Ik = θ is deterministi is not appropriate sine this implies that all insureds having a similar building would have exatly the same amount of damage. We propose an extension to the approah of Unanwa and MDonald (2000b) as follows. Assume that for eah φ and θ, the 100π%-onfidene interval lφ,θ, u φ,θ is alibrated (quantile mathing) to a beta distributed r.v. V φ,θ suh that Pr V φ,θ u φ,θ = 1 π Pr V φ,θ l φ,θ = π 2. 2 One an interpret V φ,θ as the proportion of damage for a typial building of ategory φ during a hurriane of intensity θ. Given that a building i with a low RRI i should typially suffer less damage than a high RRI building, we define U i,k Ik as Ui,k Ik = θ = V φ,θ 1.5 RRIi, (7) for RRI i [0, 1]. 5. Numerial example In this setion, we investigate hurriane risk with the models that we have presented. We first illustrate the effets of the design of a building, the different materials used and the harateristis of the insured property on the potential damage. We then present the hurriane frequeny and intensity data and the results of fitting the various models. Finally, we analyze the long-term risk management impliations of the models Impat of the building struture Suppose we have five different residential houses (φ = 1, i.e. 1 3 story residential), eah built with different materials i.e. different RRIs. The five buildings that will be used in this example are presented in Table 3. Now suppose winds of 58.1 m/s (hurriane on the limits of ategories 3 and 4 on the Saffir Simpson sale) hit eah one of the five residential buildings of Table 3 and we want to ompare the resulting.d.f. of proportions of damage (see (7)). The.d.f. is hene Vφ,θ 1.5 RRIi u Pr U i,k u Ik = θ = Pr = Pr V φ,θ u (1.5 RRI i) 1, where V φ,θ has a beta distribution with parameters 38.2 and Those parameters were obtained by alibrating a beta distribution Table 3 Different qualities of residential homes and their RRI. Name Charateristis RRI Best ase Best of all materials and riteria Worst ase Worst of all materials and riteria Example Example-N Example-I Same building as in Table 4 of Unanwa and MDonald (2000b) N for newer. Same as Example but: built within 5 years, meets ANSI/ASCE standards best envelope maintenane. I for improved. Same as Example but: roof struture is made of flat onrete tiles, windows glass is fully tempered Fig. 4. Cumulative distribution funtion of the proportions of damage beause of winds of 58.1 m/s. with the onfidene bounds for a building φ = 1 with winds of 58.1 m/s using Unanwa et al. (2000a). The different.d.f. urves for eah of the five buildings are presented in Fig. 4. We notie major differenes between the best and worst buildings. For a $ building, the probability of getting more than $ of damage given that winds of 58.1 m/s hit the building is almost 90% with the Worst ase and is 0% with the Best ase building. Moreover, the median loss with the latter building is $ while it is about $ for the former; this is twie more damage. Differenes are less important with ordinary buildings (the 3 houses having the Example prefix). We might also be interested in omparing the effets of slightly improving the onstrution of a typial residential building if winds of suh strength our. To meet that goal, we ompare the typial building Example with Example - I in whih the standard asphalt shingles roof struture is replaed with flat onrete tiles, and standard annealed windows are replaed with fully tempered windows. We obtain that the probability of getting a loss over $ is slightly less than 80% with the typial building ompared to 45% with the improved building. Median losses are approximately $ with building Example - I and $ with building Example. Although there are slight distributional differenes between the group of buildings Example, what we might learn from the best and worst buildings is that it is important for an insurer to make sure that the least number of weak omponents are attahed to the insured building. In this ase, the effets are notieable and endangers the insurability Dataset We now present the dataset that will be used for fitting the joint frequeny and intensity models. The dataset has been built

8 M. Boudreault et al. / Insurane: Mathematis and Eonomis 54 (2014) Fig. 5. Number of hurrianes that made landfall in Florida, USA, from 1899 to Fig. 6. Cumulative number of events in the dataset as a funtion of the expeted umulative number of events in model #1. Table 4 Parameters of model #1. Parameters p (0.188) λ (0.1766) p (0.2204) λ (0.2606) q (0.064) LL: q (0.0548) using two important soures. First, hurrianes from 1899 to 1998 are provided by Neumann et al. (1999) where eah hurriane is indexed aording to its month of ourrene and its intensity in eah US state it has made landfall. Sine the year 2004 has been exeptional in Florida for both ourrenes and reported losses due to hurrianes, the dataset had to be updated to inlude hurrianes from 1999 to 2004 using the Weather Underground s Hurriane Arhive. A tropial ylone had to meet the following riteria in order to be inluded in the dataset: landfall in Florida with an intensity of at least 1 or move near Florida oasts with winds felt of at least intensity 1. The resulting dataset has 68 hurrianes. Fig. 5 shows the number of hurrianes that ourred in eah year from 1899 to Fit assessment Three models with a Markovian environment In this setion, we estimate models #1, 2 and 3 (see Table 2) using maximum likelihood estimation. Annual data have been used in models #1 and 2, while monthly data were neessary in model #3. Joint estimation of frequeny and intensity models is done using hurriane ount data, along with the number of hurrianes observed in eah ategory. Note that given the total number of hurrianes within a time period, the joint distribution of the number of hurrianes within eah ategory is multinomial. We refer to Hamilton (1989), Hardy (2001) and Lu and Garrido (2006) for the estimation of Markov-swithing models. We begin with the estimation of model #1. The results are shown in Table 4 with standard errors given in parentheses. We denote by p 02 the transition probability of going from El Niño to La Niña in a year, while p 02 is the onverse. The probabilities q 0 and q 2 are the binomial probabilities in eah state. Finally, λ 0 (λ 2 ) is the mean number of events in El Niño (La Niña). LL refers to the log-likelihood obtained. We have also plotted the observed umulative number of events as a funtion of the expeted umulative number of events in model #1. If the fit is good, we expet the data points to form a 45 line. We dedue from Fig. 6 that the fit of model #1 is relatively good despite the deviations observed in the late 1940s. Table 5 Parameters of model #2. Parameters Fig. 7. Conditional intensity distribution in model #1. p 01o (0.2804) λ (0.2251) q (0.0463) p 1o (0.4523) λ (0.0924) q 1 1.4E 06 (0.0001) p 1a (0.3935) λ (0.1151) q (0.037) p 21a (0.0757) Log-likelihood: The disparity in the frequeny and intensity parameters within eah regime (q 0 vs q 2 and λ 0 vs λ 2 ) shows that there are two states of oean temperatures: one in whih both frequeny and intensity are higher, and the opposite state. Indeed, the mean number of events goes from 0.37 (El Niño) to 0.89 (La Niña), while the mean intensity (on the Saffir Simpson sale) goes from ( ) = in El Niño to ( ) = 2.58 in La Niña. The empirial observation of the effets of ENSO are thus onfirmed by the models. Fig. 7 illustrates the intensity distribution during both El Niño and La Niña. The parameters obtained with model #2 are shown in Table 5. The third neutral state added in the model is one that is low in terms of frequeny and intensity but might bring heavy rainfall. Empirial studies in meteorology (see e.g. Bove et al. (1998) and Tartaglione et al. (2003)) seem to point out that neutral ENSO phases generate more hurrianes than during El Niño years but less than during La Niña years. This has been verified on hurrianes that made landfall in the US (Bove et al., 1998) and in the Caribbean (Tartaglione et al., 2003). Their results are fairly different than those presented in Table 5 for several reasons. First, in both papers presented, the states of oean temperatures are learly

9 130 M. Boudreault et al. / Insurane: Mathematis and Eonomis 54 (2014) Table 6 Parameters of model #3. Parameters p (0.1684) λ (L) (0.5928) p (0.2082) λ (L) (1.0028) q (0.0657) α A (0.6119) q (0.0531) β A (0.4332) LL: defined using meteorologial instruments and this is not the ase with a Markov-swithing model where the label of eah state is less learly defined. Seond, both artiles provide probabilities of landfall during eah phase and annot be ompared with our work sine the territory overed with our data is smaller than the whole U.S. oasts or the Caribbean. If we also ompare the rates of arrival of hurrianes with both models #1 and #2, we observe that El Niño has more events in model #2 and less events in La Niña than model #1. This implies that ENSO phases are defined differently in model #2, making the other parameters ompensate. Finally, the effets of an ENSO-neutral phase on the intensity of hurrianes are less doumented in the meteorologial literature. Table 6 shows the parameters obtained by estimating model #3. Note that parameters λ (L) 0 and λ (L) 2 are the values of λ (L) (t) in eah state. Estimation of this model supports that during La Niña, hurrianes are more frequent (λ (L) 2 > λ (L) 0 ) and severe (q 2 > q 0 ) Doubly periodi model We have estimated with monthly data the parameters of model #4. These are a, b, α A, β A, α L, β L and also the parameters of the f funtion that transforms λ (L) (t) into a probability in the binomial intensity model. The f funtions used are the exponential.d.f. (i.e. 1 exp ( x/µ)), log logisti.d.f. (i.e. 1 (1 + x/µ) 1 ), Weibull.d.f. (i.e. 1 exp x µ σ ) and normal.d.f. (with mean µ and variane σ 2 ). Moreover, as in Lu and Garrido (2005), we have fixed the length of any yle to = 5 years and the starting point m L has been set to 3.75 years. The latter is beause their dataset and ours both start in The results are shown in Table 7. The graph of the umulative number of events observed in the dataset as a funtion of the expeted number of events in the model (see Lu and Garrido, 2005, for the expression) is shown in Fig. 8. We observe that the double-beta periodi model has an adequate overall fit but also fails to explain the abnormally large number of events in the late 1940s. Adjusting the length of the yle might improve the fit during this time period but at the prie of worsening the fit for the other time periods Comparison of models Table 8 shows the log-likelihood of eah joint frequeny and intensity model that has been estimated previously. Note that * indiates that a log logisti funtion has been piked to transform λ (L) (t) to a binomial probability (for parsimony reasons). Fig. 8. Cumulative number of events in the dataset as a funtion of the expeted umulative number of events in model #4. Table 8 Summary of the fit of the joint frequeny and intensity models. Model Log-likelihood Data freq. Benhmark Annual # Annual # Annual # Monthly # Monthly We have also estimated a benhmark model, that is taken to be a simple homogeneous Poisson proess along with an independent binomial distribution for hurriane intensity. In this model that ignores the effets of ENSO, the mean rate of arrival is hurriane per year and the probability parameter in the binomial distribution is The standard errors of these estimations are respetively and As a first step, we ompare models that have been estimated with annual data. Using the likelihood ratio test (LRT), we an ompare the benhmark model to model #1 sine the former is a speial ase of the latter. With 4 additional parameters (degrees of freedom), one gets a p-value of 6.79%. Thus, onsidering the added omplexity, model #1 has a signifiant better fit at a level of 10% but not at 5%. As seen in the plot of Fig. 6, the failure to fit the inreased hurriane ativity in the 1940s by model #1 may very well explain why the LRT has suh a p-value. Model #1 ould still be further analyzed sine (1) we annot rejet it at a level of 10%, (2) there is an interesting disparity between the parameters in eah state and (3) this will have obvious impats on the distribution of losses (see Setions 5.4 and 5.5). We further ontinue our analysis of the fit of the models by heking the appropriateness of a third state, i.e. by omparing models #1 and #2. The purpose of this test is to statistially verify the presene of another state in the ENSO phenomenon. One noties an important log-likelihood differene, i.e With 4 additional parameters with respet to model #1, the LRT yields a Table 7 Parameters of model #4. Parameters/funtion f Exponential Log logisti Weibull Normal a (0.4655) (0.4574) (0.5047) (0.3653) b (0.6159) (0.6362) (0.648) (0.516) α A (0.6157) (0.6162) (0.6051) (0.462) β A (0.4367) (0.4366) (0.429) (0.3195) α L (2.5588) (2.269) (1.6833) (0.5296) β L (2.1118) (1.9033) (1.4604) (0.5571) µ (1.334) (1.1665) ( ) (1.2762) σ (0.3052) (2.2433) Log-likelihood

10 M. Boudreault et al. / Insurane: Mathematis and Eonomis 54 (2014) Table 9 Expeted hurriane losses in the next t years for all models given that the proess starts in La Niña. t Model #1 #2 #4 Benh % % % % #1 Benh. Benh. Table 11 99% Value-at-Risk of hurriane losses in the next t years for all models given that the proess starts in La Niña. t Model #1 #2 #4 Benh % % % % #1 Benh. Benh. Table 10 Standard deviation of hurriane losses in the next t years for all models given that the proess starts in La Niña. t Model #1 #2 #4 Benh % % % % #1 Benh. Benh. Table 12 99% Tail Value-at-Risk of hurriane losses in the next t years for all models given that the proess starts in La Niña. t Model #1 #2 #4 Benh % % % % #1 Benh. Benh. p-value of Thus, a 3-state ENSO model is more appropriate for the evolution of oean temperatures. Finally, we an ompare models that use monthly data, i.e. models #3 and #4. One an see that aounting for parsimony (say using the Akaike or Bayes Information Criteria), model #3 has a muh better fit than model #4, further indiating the neessity of a Markov-swithing framework to represent both hurriane frequeny and intensity Assessing individual losses In this part of the numerial example, we apply the models presented above to evaluate the distribution of individual losses for a building loated in a hurriane-prone area. We onsider the benhmark model along with models #1, #2 and #4. The reason why we have exluded model #3 is beause for long-term risk management purposes, one does not need the exat timing of events within a year. Suppose that the insured property is worth $ and is a 1 3 story building having the harateristis of Example-N of Setion 5.1. The property is built somewhere in Florida suh that all hurrianes that our in this state will ause damage. This will overstate the risk measures and the results an be made more realisti by either using an attenuation funtion or use more loalized hurriane data (speifi to a region instead of the whole state of Florida). We also assume that the state of oean temperatures is known as being in La Niña. This is to emphasize the importane of ENSO on the values of risk measures. The following results have been omputed using simulations and are presented in what follows. Tables 9 12 show respetively the values of the expeted losses, the standard deviation, the Value-at-Risk (at a level of 99%) and the Tail VaR (also at a level of 99%) of X (t) for various models and values of t. The risk measures indiate that ignoring the dependene between hurriane frequeny and intensity may have a signifiant impat on these quantities. The relative differene between the independent (benhmark) model and the proposed models is always as large as 25% 30%, and an be muh larger in the short run. One also noties that risk measures with the doublyperiodi model is somewhere in between the Markov-swithing models and the benhmark model. This is mainly beause the length of a yle is stohasti in the proposed Markovian models, and fixed in the doubly-periodi model. This further inreases the damage potential during La Niña, rendering the doubly-periodi model less appropriate for long-term risk management purposes Solveny of the insurane ompany Suppose that a portfolio omposed of 1000 insureds is exposed to a ommon hurriane risk. Moreover, the portfolio is suh that eah property share the same insurane harateristis of the building used in Setion 5.4. Furthermore, we assume that the size and harateristis of the portfolio remain stable through time. In this example, we analyze the ruin probability of an insurer that ignores the dependene between hurriane frequeny and intensity. We denote the surplus proess of the insurane ompany by U = {U (t), t 0} where the surplus level at time t is U (t) = u + πt S (t) and S (t) is as defined in (3). Based upon the results of Setion 5.4, we assume the premium inome per unit of time π to be equal to 1.25 times the expeted annual losses in the independent model and the initial surplus to be twie as large as the expeted annual losses in that model, meaning π = 1.25 E [S BENCH (1)] and U (0) = u = 2 E [S BENCH (1)]. We denote by the rv τ the time of ruin where inf {s, U (s) < 0}, if U goes below 0 at least one τ = s>0, if U never goes below 0. The finite-time ruin probability over (0, t] is given by ψ (u, t) = Pr (τ t U (0) = u). We use stohasti simulations to approximate ψ (u, t). Fig. 9 depits the values of the ruin probability ψ (u, t) under eah model for t (0, 30], given that the state of oean temperature is in La Niña. For all three models, ignoring dependene between hurriane frequeny and intensity signifiantly aggravates the solveny of the insurane portfolio, espeially after approximately eight years. For example, the 15-year ruin probability is about 65% in model #1 while it is about 40% in the benhmark model. Thus, the flow of premiums and the surplus should aount for the state of ENSO to ensure the solveny of the portfolio. 6. Conluding remarks In the meteorologial literature, it is doumented that various phenomena (like El Niño/Southern Osillation (ENSO) and the Atlanti Multideadal Osillation) influene hurriane frequeny and intensity. In this paper, we have adapted models with a Markovian environment to hurriane losses that aount for these meteorologial proesses in a risk theory ontext. Moreover, we have introdued frameworks for hurriane frequeny, intensity and damage. The results that we obtain with Markovian models onfirm the existene of a low frequeny/low intensity and a high

11 132 M. Boudreault et al. / Insurane: Mathematis and Eonomis 54 (2014) Fig. 9. Ruin probability of the insurer when premiums and surplus are based upon independene between frequeny and intensity of hurrianes. frequeny/high intensity state, that we have interpreted as being El Niño and La Niña respetively. This has an important effet in risk management as illustrated by the various risk measures and solveny results. We found the doubly-periodi model of Lu and Garrido (2005) to be less appropriate for risk management purposes beause the duration of stay in La Niña is deterministi. One may believe that hurriane losses seem to reah new reord highs every year due to global warming. However, studies by Pielke and Landsea (1998) and Pielke (2005) state that with normalized data (data orreted for inflation, hanges in population and wealth), losses in the 1990s were lower than in any other deade. (Knutson et al. 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