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Insurance: Mathematics and Economics 54 (2014) 123–132

Contents lists available at ScienceDirect

Insurance: Mathematics and Economics

journal homepage: www.elsevier.com/locate/ime

Risk models with dependence between claim occurrences andseverities for Atlantic hurricanes

Mathieu Boudreault a,∗, Hélène Cossette b, Étienne Marceau ba Université du Québec à Montréal, Département de mathématiques, C.P. 8888, succ. Centre-Ville, Montréal, QC, H3C 3P8, Canadab Université Laval, École d’actuariat, Québec, QC, G1V 0A6, Canada

a r t i c l e i n f o

Article history:Received November 2010Received in revised formSeptember 2013Accepted 5 November 2013

Keywords:Risk theoryHurricane riskRisk measuresEl Niño/Southern Oscillation (ENSO)Florida hurricanes

a b s t r a c t

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 hurricane risk context. Thesemodels are supported by the findings that ElNiño/Southern Oscillation (as well as other natural phenomena) influence both the number of hurricanesand their strength. Hurricane risk is thus broken into three components: frequency, intensity and damagewhere the first two depend on the state of the Markov chain and intensity influences the amount ofdamage to an individual building. The proposed models are estimated with Florida hurricane data andseveral risk measures are computed over a fictitious portfolio.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Supported by natural and economic phenomena, there hasbeen a rising interest in the risk theory literature on modelslinking claim occurrence and severity. Two important classes ofdependence between the latter two components are: (1) renewalmodels inwhich the interarrival time and claim size are dependent(see for instance Albrecher and Teugels (2006), Boudreault et al.(2006), Cossette et al. (2008), Badescu et al. (2009), Cheunget al. (2010) and references therein) and (2) risk models with aMarkovian environment where interarrival times and claim sizes(and possibly premiums as well) all depend upon the state ofa common Markov chain (see for example Asmussen (1989), Luand Li (2005), Lu (2006) and Ng and Yang (2006) and referencestherein). For a general review of the latter models, see Chapter 7 inAsmussen and Albrecher (2010).

In the climatology and meteorology literature, it is well knownthat the phenomenon known as El Niño/Southern Oscillation in-fluences both the number of hurricanes and their strengths (windspeed, amount of precipitation, etc.) (see for example Gray (1984),Meyer et al. (1997) and Landsea and Pielke (1999); Pielke and Land-sea (1998) among others). A dependence relationship between

∗ Corresponding author. Tel.: +1 514 987 3000; fax: +1 514 987 8935.E-mail addresses: [email protected] (M. Boudreault),

[email protected] (H. Cossette), [email protected](É. Marceau).

hurricane frequency and intensity is thus obvious allowingMarko-vian models cited above to be adapted and refined to suit such ahurricane context. Note that in all the aforementioned risk theorypapers, the focus has been put mostly on deriving ruin measures.In this paper, we extend the previous class of Markovianmodels inthe line of Cossette et al. (2003) by decomposing natural catastro-phe risk into frequency, intensity (strength of the event) and dam-age.

Based upon the literature in climatology and meteorology, wepropose different joint hurricane frequency and intensity models.We represent by a latent process the current state of hurricaneactivity, which can be influenced by many physical phenomena.We first introduce a Markov-switching framework where bothfrequency and intensity are allowed to be state-dependent.Modelswith two states or three states are considered. We also extendthe frequency models of Lu and Garrido (2006, 2005) such thathurricane frequency and intensity are dependent. Using Floridalandfalling hurricane data (both frequency and intensity) andcivil engineering approaches to quantify damage, we estimateand compare the latter models in order to analyze various riskmeasures of a fictitious portfolio of policyholders.

The paper is structured as follows. Section 2 introduces thegeneral modeling framework for hurricane risk. In Section 3 wedetail the joint frequency and intensity models proposed, whileSection 4 focuses on the damage component. In Section 5, we applythe models to Florida data. Finally, Section 6 ends the paper with aconclusion.

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124 M. Boudreault et al. / Insurance: Mathematics and Economics 54 (2014) 123–132

Table 1Saffir–Simpson hurricane intensity scale.

Category Description MSWS (m/s) Past count

1 Minimal 33–42 222 Moderate 43–49 183 Extensive 50–58 204 Extreme 59–69 75 Catastrophic over 69 1

2. Modeling hurricane risk

2.1. Introduction

According to the National Hurricane Center (of the NationalOceanic and Atmospheric Administration (NOAA)) and to FederalEmergencyManagementAgency (FEMA), a hurricane is ‘‘[. . . ] a typeof tropical cyclone, the generic term for a low pressure system thatgenerally forms in the tropics. A typical cyclone is accompanied bythunderstorms, and in the Northern Hemisphere, a counterclock-wise circulation of winds near the earth’s surface.’’ Depending onthe tropical cyclone’s location or strength, a tropical cyclone maybe known as a hurricane, typhoon, tropical stormor depression (formore details, see Neumann (1993)).

Hurricane activity in the Atlantic Ocean and on the AmericanEast Coast is known to be influenced by many phenomena suchas the Atlantic Multidecadal Oscillation (AMO) (see Chylek andLesins (2008), and the NOAA Frequently Asked Questions), ElNiño/Southern Oscillation (ENSO) (see Gray (1984), Meyer et al.(1997) and Landsea and Pielke (1999); Pielke and Landsea (1998)among others) and climate change (see Emanuel (2005) andWMO(2006)).

For example, ENSO represents the cyclical patterns observed inthe surface temperature of the Pacific Ocean and its changes inair surface pressure. During cycles that may last months or years,ocean temperatures in the central tropical Pacific Ocean tend towarm (El Niño) and then cool (La Niña) in a cyclical pattern. Ac-companying these temperature variations are changes in air sur-face pressure across the Pacific. During El Niño (La Niña), higher(lower) pressures are observed in the Western Pacific and lower(higher) pressures are observed in the Eastern part of the basin.The cyclical changes in air pressure is known as the El Niño South-ern Oscillation (ENSO). Many authors have reported that ENSO isknown to have an important influence on hurricane frequency andintensity in the Atlantic Ocean (see aforementioned authors). In-deed, during La Niña, more hurricanes are generated (on average)and these hurricanes are generally stronger. This is the usually theopposite in El Niño. This means that frequency and intensity aredependent components of hurricane risk. Note that frequency rep-resents the number of hurricanes that made landfall in a given re-gion within a specific time period (year, month).

Intensity is defined as the strength of a hurricane at a givenlocation and is generally measured on the Saffir–Simpson scale,which is based upon wind speeds. The latter classifies hurricanesaccording to five levels (see Table 1 for a description), whichare distinguished on the basis of the 1-min maximum sustainedwind speed (MSWS). We mention that instruments measure theaverage wind speed within 1-min time intervals. The MSWS is thehighest of these means. Moreover, the approach is not limited tothe Saffir–Simpson scale. One may also include less severe tropicalstorms or classify hurricanes in more categories based on theMSWS or other measures. Whenwinds are between 18 and 32m/s(meters per second), the cyclone is classified as a tropical stormand below 17m/s, the storm is a depression. In the latter cases, theBeaufort scale is used. In the fourth column of Table 1, we indicatethe distribution of hurricane intensity, for hurricanes that madelandfall in Florida. The dataset used to compute the numbers in thiscolumn is discussed in Section 5.

Finally, damage is related to the amount of losses suffered by apolicyholder for a givenhurricane. This component is closely linkedto the intensity of the hurricane and is presented in more detail inSection 4.

2.2. General modeling framework

Let N = {N (t) , t > 0} represent the counting process of thenumber of hurricanes that make landfall in a given region duringthe time interval (0, t]. Let also the random variable (r.v.) Ik rep-resent the intensity of the k-th hurricane, which is, as discussedearlier, the strength of a given hurricane on a given scale. We alsodefine the process X i = {Xi (t) , t > 0} where Xi (t) is the totalamount of losses suffered by policyholder i due to the N (t) hurri-canes that occurred in (0, t] i.e.

Xi (t) =

N(t)k=1

Ci,k, N (t) > 0,

0, N (t) = 0.(1)

The amount of loss due to the k-th hurricane is defined by the r.v.Ci,k with

Ci,k = Ui,k × bi, (2)

where the scalar bi is the exposure or the value of the insured build-ing and the r.v. Ui,k ∈ [0, 1] represents the proportion of damage.The information regarding the type of building and its constructionwill be embedded in Ui,k. Moreover, the intensity of the k-th hurri-cane will influence the extent of damage to a property so that theconditional distribution of Ui,k depends upon the intensity r.v. Ik.The specific relationship between Ui,k and Ik will be defined laterin Section 4.

The way that we define Ci,k assumes that losses to an individualbuilding cannot be larger than its given value bi, or in other words,Ci,k ∈ [0, bi] . The type of building and the force of the hurricanewill determine the distribution of Ci,k and thus the total losssuffered by policyholder i for hurricane k.

For a portfolio of n policyholders living in a hurricane-pronearea, the process for the aggregate losses is defined by S ={S (t) , t > 0} where S (t) is the aggregate losses for the timeperiod (0, t] e.g.

S (t) =n

i=1

Xi (t) =

N(t)k=1

ni=1

Ci,k, N (t) > 0,

0, N (t) = 0.(3)

We interpretn

i=1 Ci,k as the aggregate amount of losses due tothe kth hurricane. There are two sources of dependence withinthismodel. First, the number of hurricanes and their intensities arecommon to all policyholders of the same region. Second, as men-tioned in the Introduction, ENSO induces a dependence relation be-tween hurricane frequency and intensity. This is detailed next.

2.3. El Niño/Southern oscillation

As previously mentioned in Section 2.1, various phenomenasuch as AMO, ENSO and climate change influence the hurricaneactivity level. Although ENSO is observed via the Oceanic Niño(ONI) and Southern Oscillation Indices (SOI), the fact that manymeteorological phenomena interact to influence the hurricane ac-tivity level (not just ENSO) justifies the use of a latent process ap-proach. However, to lighten the presentation, we will interpretthe latent stochastic process as ENSO with states correspondingroughly to El Niño and La Niña even though there might not bean exact correspondence with the ONI and SOI. Thus, the interpre-tations that we attribute to the states of ENSO can be described


M. Boudreault et al. / Insurance: Mathematics and Economics 54 (2014) 123–132 125

Fig. 1. Graphical summary of the various components of the model.

as (low frequency, low intensity)/(high frequency, high intensity)states or El Niño/La Niña respectively.

Let R = {R (t) , t ≥ 0} be the process that represents the timelyevolution of ENSO and I = {I1, I2, . . .} be a sequence of r.v.s whereIk is the intensity of the kth hurricane. In Section 3, we definespecificmodels for R,N and I where R has a simultaneous influenceon the latter two. Fig. 1 illustrates the various components of themodel and their interactions.

Moreover,we suppose that given ENSO, frequency and intensityare independent and time-independent. In other words, once weknow the state of ENSO at time t , the intensity of hurricanesthat occurred within a period is independent from the number ofhurricanes that happened during the same period. Furthermore,once the evolution of ENSO is known, frequency and intensity ofhurricanes are serially-independent. Finally, we assumeminimumcollateral damage meaning that if two neighbor buildings sufferdamage, it is because they were exposed to the same hurricane.This means that we exclude possibilities that may create furtherdependence between buildings, apart from the common exposureto ENSO. Mathematically, this means that given ENSO, the numberof hurricanes and the intensity Ik of a given hurricane, the r.v.sU1,k,U2,k, . . . ,Un,k are conditionally independent.

3. Joint frequency and intensity models

In this section we consider three models with a Markovianenvironment and adoubly periodicmodel to represent ENSO, alongwith models for the frequency and intensity of hurricanes.

3.1. Three models with a Markovian environment

Within the three models with a Markovian environment, R isassumed to be a latent discrete-time Markov chain where eachstate defines the status of ENSO. Because ENSO is a long-termphenomenon (duration of several years for example), transitionsof R between states are assumed annual unless stated otherwise.

Furthermore, in everything that follows, we assume that giventhe state of R, the conditional distribution of the intensity of the

k-th hurricane is binomial. Hence, because we use the Saffir–Simpson scale on values {1, 2, 3, 4, 5}, we have that

( Ik| R (t))− 1 ∼ Binom4, qR(t)

, (k = 1, 2, . . .),

where the probability parameter qR(t) evolves with R. We expectqR(t) to be higher (lower) during La Niña (El Niño) so that severehurricanes (4 and 5) will be more (less) likely.

We present three different frequency models: (1) a two-state Markov-switching Poisson process, (2) a three-state Markov-switching Poisson process and (3) a two-state non-homogeneousPoisson process.

3.1.1. Two-state Markov-switching Poisson processIn the two-state Markov-switching Poisson process, we sup-

pose that R is a latent two-state Markov chain such that R (t) = 0represents El Niño and R (t) = 2 denotes La Niña. Thus, hurricanefrequency is a Poisson process such that the rate of arrival of hur-ricanes at time t , denoted by λR(t), depends on the value of R (t) .

3.1.2. Three-state Markov-switching Poisson processIn the meteorology literature, a third state of ocean tempera-

tures has been considered in e.g. Landsea and Pielke (1999) andKatz (2002, 2008). This is known as a neutral state, that occurs be-tween transitions from La Niña to El Niño and vice-versa. Then,to take into account this phenomenon, we propose in the secondmodel with a Markovian environment that R to be a latent Markovchain that follows a cyclical pattern illustrated in Fig. 2.

Note that one cannot use a standard three state Markov chainbecause of this type of cyclicality. Indeed, we cannot observe anobservation of El Niño followed by Neutral and then El Niño. Oncein a Neutral state, ENSO will eventually return to a La Niña phase.To accomplish this in a Markovian environment, we define a fourstate Markov model, with El Niño and La Niña states, and twoNeutral states. However, the two Neutral states will have the sameconditional frequency and intensity. We denote by R (t) = 1othe neutral state of ocean temperatures at time t given that atsome time in the past, the transition to Neutral came from El Niño.Similarly, we define R (t) = 1a to be the neutral state of surfacewater temperatures at time t given at some time in the past, the



Fig. 2. Illustration of the evolution between the states in the three-state Markov-switching process.

transition toNeutral came fromLaNiña. Thus, the transitionmatrixP has 4 dimensions and becomes

P =

p00 p01o 0 00 p1o1o 0 p1o2p1a0 0 p1a1a 00 0 p21a p22

. (4)This transition matrix can be interpreted as follows. From El Niño,ocean temperatures either remain in El Niño or switches to aneutral phase. Once in a neutral phase, R cannot switch back toEl Niño and is forced to either stay in a neutral phase or moveto La Niña. This means that a move from El Niño to La Niña orvice versa must be made using a transition to a neutral phase.This formulation of the ENSO cycle is technically a 4-state Markovchain, but states 1o and 1a are both considered the neutral state. Inthis framework, we assume that hurricane frequency is a Markov-switching Poisson process with 3 states (since λ1o = λ1a) andintensity also relies on 3 different states.

3.1.3. Two-state non-homogeneous Poisson processIn Lu and Garrido (2006), R is also a two-state latent Markov

chain. However, given the state of R, hurricanes arrive according toa non-homogeneous Poisson process. In their approach, R evolveson an annual basis, but the non-homogeneous Poisson process cap-tures the fact that hurricanes occur between June and November,which is known as the hurricane season. They define an annual pe-riodicity function λ(A) (t) (t ∈ [0, 1]) such that

λ(A) (t) ≡

0, 0 ≤ t <

512,

kT (t)αA−1 1 − T (t)βA−1 , 5

12≤ t ≤

1112,

0,1112< t ≤ 1,

(5)

where

T (t) = 126

t −

512

k−1 =

T t∗A αA−1 1 − T t∗A βA−1t∗A =

512

+612

αA − 1αA + βA − 2

.

The parameters αA and βA will determine the shape of λ(A) (t)i.e. when hurricanes are more likely to occur within a year. More-over, t∗A is the moment when λ

(A) (t) reaches its mode, such thatλ(A)

t∗A

= 1. Because the annual periodicity function λ(A) (t)

varies within [0, 1], Lu and Garrido (2006) define the rate of arrivalof hurricanes as

λ (t) = λ(L) (t)× λ(A) (t − ⌊t⌋) , t ≥ 0,

where ⌊t⌋ is the floor function. The function λ(L) (t), defined as

λ(L) (t) =

λ(L)1 , R (t) = 1

λ(L)2 , R (t) = 2,

scales λ(A) (t) to get a representative rate of arrival function. Weuse the notation λ(A) and λ(L) to contrast between the annual andlong-term components of the rate of arrival of hurricanes.

3.2. Doubly periodic model

Lu and Garrido (2005) have proposed a doubly periodic non-homogeneous Poisson process for hurricane frequency. We com-bine this frequency model with an intensity model to account fordependence between these two components of hurricane risk. Wefirst start by briefly summarizing their approach. Note that in thedoubly periodic model, R is a deterministic process.

In the doubly periodic model, the short-term rate of arrival ofhurricanes λ(A) (t) defined in (5) is affected by R, which is in fact adeterministic periodic function representing ENSO. The annual rateof arrival of hurricanes λ(A) (t) is multiplied by a constant whichevolves periodically, according to smooth transitions of R betweenEl Niño and La Niña. Consequently, the intensity function λ (t) issuch that

λ (t) = λ(L)

⌊t⌋ − ctc

+ t∗A

× λ(A) (t − ⌊t⌋) , t > 0,

where R (t) = λ(L) (t) is the long-term intensity function whichcharacterizes ENSO, c (c = 1, 2, 3, . . .) is the length of an ENSOcycle, and λ(L) (t) is defined as

λ(L) (t) = a +b − akL

t − mL

c−

t − mL

c

αL−1×

1 −

t − mL

c−

t − mL

c

βL−1for t > 0, where

kL =t∗L − mL

c

αL−1 1 −

t∗L − mLc

βL−1and

t∗L = mL + c

αL − 1αL + βL − 2

.

Moreover, t∗L is the mode of λ(L) (t), kL is a scaling factor, a (b)

represents the minimum (maximum) amplitude of peak valuesand mL represents the time when the lowest point is reached byλ(L) (t) (described by Lu and Garrido (2005) as the starting pointof the complete long-term cycle). The minimum a (maximum b)amplitude of peak values corresponds to El Niño (La Niña). Notethat αL and βL determine the shape of the periodicity embedded inλ(L) (t), just like αA and βA do for λ(A) (t) .

Fig. 3 illustrates the path of the deterministic process R definedby the doubly-periodic function, in the context of counting thenumber of hurricanes over 10 years (from Lu and Garrido (2005)).The ‘‘short-term’’ curve provides the rate of arrival of hurricanesover each year, i.e. λ (t) and emphasizes that they only occurbetween June andNovember. The peak rate of arrival is shifted by afactor (‘‘long-term’’ curve or the function λ(L)) that evolves slowlyover the long-term, i.e. ENSO.

Lu and Garrido (2005) solely focus on hurricane frequency butone needs to relate the intensity with ENSO, which is determinedby λ(L) (t). To do so, we need to transform this [0,∞[ input intoa valid parameter for the binomial distribution. This can easily bedone using a transformation f : [0,∞[ → [0, 1] such as thecumulative distribution function (c.d.f.) of a positive r.v. Thus, ourextension to account for intensity is such that Ik follows a binomial



Fig. 3. Illustration of a doubly periodic rate of arrival function.Source: Taken directly from Lu and Garrido (2005)’s Figure 2.

distribution over {1, 2, 3, 4, 5} with probability

qt = fλ(L) (t)

.

This will be illustrated in the numerical example of Section 5.Finally, Lu and Garrido (2005) only represented hurricanes

within a year and over cycles that last approximately 3–5 years.One can also use the approach of Lu and Garrido (2005) to modelAMO, which can last decades, and ENSO, which may last a fewyears.

3.3. Summary

We provide in Table 2 a summary of all the joint frequency andintensity models that we have proposed. Note that M.-S. stands forMarkov-switching.

4. Modeling hurricane damage

4.1. Introduction

According to the FEMA, damage from hurricanes mainly comesfrom high winds and heavy rain but for buildings built alongcoastal areas, storm surges are severe threats. A storm surge is anabnormal rise of the sea level due to high winds, that will resultin important floods. Damage to buildings can range from brokenwindows to collapsing roofs or quasi-total destruction (since thefoundations will remain). For example, the National Oceanic andAtmospheric Administration (NOAA) describes the damage forframe homes resulting from a category 4 hurricane as:

‘‘Poorly constructed homes can sustain complete collapse of allwalls as well as the loss of the roof structure. Well-built homesalso can sustain severe damage with loss of most of the roofstructure and/or some exterior walls. Extensive damage to roofcoverings, windows, and doors will occur. Large amounts ofwindborne debris will be lofted into the air. Windborne debrisdamage will break most unprotected windows and penetratesome protected windows.’’

The damage model considered in this paper directly linksdamage towind speed and is based uponUnanwa et al. (2000a) andUnanwa and McDonald (2000b). Indirect damage that may comefromheavy rain into a damaged building (from the roof and brokenwindows) is also taken into account. Thus, water infiltration fromthe ground is not accounted for, which excludes damage fromstorm surges. The types of damage accounted for in these twopapers are consistent with homeowners insurance since floods are

Table 2Summary of the joint frequency and intensity models proposed.

Model ENSO Frequency Intensity

#1 2-stateMarkov chain

M.-S. Poisson process M.-S. binomialdistribution


M.-S. Poisson process M.-S. binomialdistribution


M.-S. non-homo.Poisson process

M.-S. binomialdistribution

#4 Deterministicprocess

Non-homo. Poissonprocess

Binomialdistributionqt = f

λ(L) (t)

not usually covered by insurance companies, but water that getsinto the property from the roof and brokenwindows resulting fromdamage caused by high winds may be covered. According to the2011 Florida Statutes, Title XXXVII, Chapter 627 and Section 4025,Paragraph (2)(a):

‘‘Hurricane coverage’’ is coverage for loss or damage caused bythe peril of windstorm during a hurricane. The term includesensuing damage to the interior of a building, or to propertyinside a building, caused by rain, snow, sleet, hail, sand, or dustif the direct force of the windstorm first damages the building,causing an opening through which rain, snow, sleet, hail, sand,or dust enters and causes damage.

4.2. Wind damage bands

The methodology developed in Unanwa et al. (2000a) relies ona list of building components thatmight fail because of highwinds.Those components are listed as:

‘‘[. . . ] roof covering, roof structure, exterior doors and windows,exterior wall (includes finishes, electrical and mechanical com-ponents supported, cladding and support systems), interior (in-clude contents),1 structural system (includes columns, girders,elevated floors and conveying equipment) and foundation.’’

Four different categories of buildings have been analyzedin Unanwa et al. (2000a): 1–3 story residential, commercial/industrial, government/institutional and 4–10 story mid-risebuildings. Confidence (wind damage) bands as a function of windspeed for each type of building are illustrated in Figs. 11–14 ofUnanwa et al. (2000a). Using the Saffir–Simpson scale (see Table 1),one deduces that for a 1–3 story residential building, the 95% con-fidence interval for proportions of damage is [7%, 30%] when thewinds are at 50 m/s, which is a light category 3 hurricane.

4.3. Individual adjustments

The wind damage bands were intended for a typical propertywithin a very wide category. However, not all residential proper-ties have been constructed like the typical 1–3 story residentialbuilding described in Unanwa et al. (2000a). The approach pre-sented in their second paper, i.e. Unanwa and McDonald (2000b),allows for very exhaustive individual adjustments. As much as 20different criteria are introduced in this paper to differentiate resi-dential buildings. Examples of criteria are roof covering, geometryand span, building code and age, types of windows glass, etc.

Many experts have been gathered to evaluate the potentialfailure of each building component to compute the global relative

1 Interior contents mean counters, cupboards, sinks, etc., i.e. things that areinside the property and are physically attached to the building. It does not includefurnitures, appliances or electronics.



resistivity index (RRI) of a building. This global index takes valuebetween 0 and 1 and it is obtained byweighting the quality of eachcomponent for a given building. Table 2 of Unanwa and McDonald(2000b) provides weights and quality factors for each possiblevalue of each component. The weighted sum of quality factors fora particular building provides its RRI . The value of RRI for a typicalbuilding is 0.5. If the value of RRI is lower (greater) than 0.5, itindicates that the building i is more (less) resistant than the typicalbuilding of its category.

Assume that policyholder i owns a property that belongs to oneof the 4 building categories (φ = 1, 2, 3, 4) with each categoryhaving a given RRIi. Then according to Unanwa and McDonald(2000b),Ui,k

Ik = θ = lφ,θ + RRIi × uφ,θ − lφ,θ (6)which is deterministic for θ = 1, 2, . . . , 5. Note that

lφ,θ , uφ,θ

is the 95% confidence interval for proportions of damage for abuilding that belongs to category φ during a hurricane of intensityθ .

The fact that Ui,k Ik = θ is deterministic is not appropriate

since this implies that all insureds having a similar building wouldhave exactly the same amount of damage. We propose an exten-sion to the approach of Unanwa andMcDonald (2000b) as follows.

Assume that for each φ and θ , the 100π%-confidence intervallφ,θ , uφ,θ

is calibrated (quantile matching) to a beta distributed

r.v. Vφ,θ such that

PrVφ,θ ≤ uφ,θ

= 1 −

π

2

PrVφ,θ ≤ lφ,θ

=π

2.

One can interpret Vφ,θ as the proportion of damage for a typicalbuilding of category φ during a hurricane of intensity θ . Given thata building iwith a low RRIi should typically suffer less damage thana high RRI building, we define Ui,k

Ik asUi,k

Ik = θ = Vφ,θ 1.5−RRIi , (7)for RRIi ∈ [0, 1].

5. Numerical example

In this section, we investigate hurricane risk with the modelsthat we have presented. We first illustrate the effects of the designof a building, the different materials used and the characteristicsof the insured property on the potential damage. We then presentthe hurricane frequency and intensity data and the results offitting the various models. Finally, we analyze the long-term riskmanagement implications of the models.

5.1. Impact of the building structure

Suppose we have five different residential houses (φ = 1, i.e.1–3 story residential), each built with different materials i.e. dif-ferent RRIs. The five buildings that will be used in this example arepresented in Table 3.

Now suppose winds of 58.1 m/s (hurricane on the limits ofcategories 3 and 4 on the Saffir–Simpson scale) hit each one of thefive residential buildings of Table 3 and we want to compare theresulting c.d.f. of proportions of damage (see (7)). The c.d.f. is hence

PrUi,k ≤ u

Ik = θ = Pr Vφ,θ 1.5−RRIi ≤ u= Pr

Vφ,θ ≤ u(1.5−RRIi)

−1,

where Vφ,θ has a beta distribution with parameters 38.2 and 56.6.Those parameters were obtained by calibrating a beta distribution

Table 3Different qualities of residential homes and their RRI.

Name Characteristics RRI

Best case Best of all materials and criteria 0.095

Worst case Worst of all materials and criteria 0.848

Example Same building as in Table 4 of Unanwa andMcDonald (2000b)

0.633

Example-N N for newer. Same as ‘‘Example’’ but: built within5 years, meets ANSI/ASCE standards bestenvelope maintenance.

0.595

Example-I I for improved. Same as ‘‘Example’’ but: roofstructure is made of flat concrete tiles, windowsglass is fully tempered.

0.477

Fig. 4. Cumulative distribution function of the proportions of damage because ofwinds of 58.1 m/s.

with the confidence bounds for a building φ = 1 with winds of58.1 m/s using Unanwa et al. (2000a).

The different c.d.f. curves for each of the five buildings arepresented in Fig. 4. We notice major differences between the bestand worst buildings. For a 100 000$ building, the probability ofgetting more than 50 000$ of damage given that winds of 58.1 m/shit the building is almost 90% with the ‘‘Worst case’’ and is 0% withthe ‘‘Best case’’ building. Moreover, the median loss with the latterbuilding is 28 000$ while it is about 56 000$ for the former; this istwice more damage. Differences are less important with ordinarybuildings (the 3 houses having the ‘‘Example’’ prefix).

Wemight also be interested in comparing the effects of slightlyimproving the construction of a typical residential building ifwinds of such strength occur. To meet that goal, we comparethe typical building ‘‘Example’’ with ‘‘Example - I’’ in which thestandard asphalt shingles roof structure is replaced with flatconcrete tiles, and standard annealed windows are replaced withfully tempered windows. We obtain that the probability of gettinga loss over 40 000$ is slightly less than 80% with the typicalbuilding compared to 45% with the improved building. Medianlosses are approximately 38 000$ with building ‘‘Example - I’’ and43 000$ with building ‘‘Example’’.

Although there are slight distributional differences between thegroup of buildings ‘‘Example’’, what we might learn from the bestand worst buildings is that it is important for an insurer to makesure that the least number of weak components are attached tothe insured building. In this case, the effects are noticeable andendangers the insurability.

5.2. Dataset

We now present the dataset that will be used for fitting thejoint frequency and intensity models. The dataset has been built



Fig. 5. Number of hurricanes thatmade landfall in Florida, USA, from 1899 to 2004.

Table 4Parameters of model #1.

Parameters

p02 0.3613 (0.188) λ0 0.3695 (0.1766)p20 0.3296 (0.2204) λ2 0.8909 (0.2606)q0 0.0689 (0.064) LL: −194.52q2 0.3950 (0.0548)

using two important sources. First, hurricanes from 1899 to 1998are provided by Neumann et al. (1999) where each hurricane isindexed according to its month of occurrence and its intensity ineach US state it has made landfall. Since the year 2004 has beenexceptional in Florida for both occurrences and reported losses dueto hurricanes, the dataset had to be updated to include hurricanesfrom 1999 to 2004 using the Weather Underground’s HurricaneArchive. A tropical cyclone had to meet the following criteria inorder to be included in the dataset: landfall in Florida with anintensity of at least 1 ormove near Florida coasts withwinds felt ofat least intensity 1. The resulting dataset has 68 hurricanes. Fig. 5shows the number of hurricanes that occurred in each year from1899 to 2004.

5.3. Fit assessment

5.3.1. Three models with a Markovian environmentIn this section, we estimate models #1, 2 and 3 (see Table 2)

using maximum likelihood estimation. Annual data have beenused in models #1 and 2, while monthly data were necessaryin model #3. Joint estimation of frequency and intensity modelsis done using hurricane count data, along with the number ofhurricanes observed in each category. Note that given the totalnumber of hurricaneswithin a time period, the joint distribution ofthe number of hurricanes within each category is multinomial. Werefer to Hamilton (1989), Hardy (2001) and Lu and Garrido (2006)for the estimation of Markov-switching models.

We begin with the estimation of model #1. The results areshown in Table 4 with standard errors given in parentheses. Wedenote by p02 the transition probability of going from El Niño to LaNiña in a year, while p02 is the converse. The probabilities q0 andq2 are the binomial probabilities in each state. Finally, λ0 (λ2) isthe mean number of events in El Niño (La Niña). LL refers to thelog-likelihood obtained.

We have also plotted the observed cumulative number ofevents as a function of the expected cumulative number of eventsin model #1. If the fit is good, we expect the data points to form a45◦ line.We deduce from Fig. 6 that the fit of model #1 is relativelygood despite the deviations observed in the late 1940s.

Fig. 6. Cumulative number of events in the dataset as a function of the expectedcumulative number of events in model #1.

Fig. 7. Conditional intensity distribution in model #1.


Parameters

p01o 0.5248 (0.2804) λ0 0.4862 (0.2251) q0 0.0619 (0.0463)p1o2 0.8463 (0.4523) λ1 0.1549 (0.0924) q1 1.4E−06 (0.0001)p1a0 0.8703 (0.3935) λ2 0.7567 (0.1151) q2 0.3792 (0.037)p21a 0.2040 (0.0757) Log-likelihood: −176.69

The disparity in the frequency and intensity parameters withineach regime (q0 vs q2 and λ0 vs λ2) shows that there are two statesof ocean temperatures: one in which both frequency and intensityare higher, and the opposite state. Indeed, the mean number ofevents goes from 0.37 (El Niño) to 0.89 (La Niña), while the meanintensity (on the Saffir–Simpson scale) goes from (0.0689 × 4 +1) = 1.2756 in El Niño to (0.3950 × 4 + 1) = 2.58 in La Niña. Theempirical observation of the effects of ENSO are thus confirmed bythe models. Fig. 7 illustrates the intensity distribution during bothEl 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 interms of frequency and intensity but might bring heavy rainfall.Empirical studies in meteorology (see e.g. Bove et al. (1998) andTartaglione et al. (2003)) seem to point out that neutral ENSOphases generatemore hurricanes than during El Niño years but lessthan during La Niña years. This has been verified on hurricanes thatmade landfall in the US (Bove et al., 1998) and in the Caribbean(Tartaglione et al., 2003). Their results are fairly different thanthose presented in Table 5 for several reasons. First, in bothpapers presented, the states of ocean temperatures are clearly




Parameters

p02 0.3419 (0.1684) λ(L)0 1.4019 (0.5928)

p20 0.3327 (0.2082) λ(L)2 3.3103 (1.0028)

q0 0.0809 (0.0657) αA 3.2012 (0.6119)q2 0.4002 (0.0531) βA 2.3156 (0.4332)LL: −287.58

defined using meteorological instruments and this is not the casewith a Markov-switching model where the label of each state isless clearly defined. Second, both articles provide probabilities oflandfall during each phase and cannot be compared with our worksince the territory covered with our data is smaller than the wholeU.S. coasts or the Caribbean. If we also compare the rates of arrivalof hurricanes with bothmodels #1 and #2, we observe that El Niñohasmore events inmodel #2 and less events in La Niña thanmodel#1. This implies that ENSO phases are defined differently in model#2, making the other parameters compensate. Finally, the effectsof an ENSO-neutral phase on the intensity of hurricanes are lessdocumented in the meteorological 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) ineach state. Estimation of this model supports that during La Niña,hurricanes are more frequent (λ(L)2 > λ

(L)0 ) and severe (q2 > q0).

5.3.2. Doubly periodic modelWehave estimatedwithmonthly data the parameters of model

#4. These are a, b, αA, βA, αL, βL and also the parameters of the ffunction that transforms λ(L) (t) into a probability in the binomialintensity model. The f functions used are the exponential c.d.f.(i.e. 1 − exp (−x/µ)), log–logistic c.d.f. (i.e. 1 − (1 + x/µ)−1),

Weibull c.d.f. (i.e. 1−exp−

xµ

σ) and normal c.d.f. (withmean

µ and variance σ 2). Moreover, as in Lu andGarrido (2005), we havefixed the length of any cycle to c = 5 years and the starting pointmL has been set to 3.75 years. The latter is because their datasetand ours both start in 1899. The results are shown in Table 7.

The graph of the cumulative number of events observed in thedataset as a function of the expected number of events in themodel (see Lu and Garrido, 2005, for the expression) is shown inFig. 8. We observe that the double-beta periodic model has anadequate overall fit but also fails to explain the abnormally largenumber of events in the late 1940s. Adjusting the length of thecycle might improve the fit during this time period but at the priceof worsening the fit for the other time periods.

5.3.3. Comparison of modelsTable 8 shows the log-likelihood of each joint frequency and

intensity model that has been estimated previously. Note that ‘‘*’’indicates that a log–logistic function has been picked to transformλ(L) (t) to a binomial probability (for parsimony reasons).

Fig. 8. Cumulative number of events in the dataset as a function of the expectedcumulative number of events in model #4.

Table 8Summary of the fit of the joint frequency and intensity models.

Model Log-likelihood Data freq.

Benchmark −198.89 Annual#1 −194.52 Annual#2 −176.69 Annual#3 −287.58 Monthly#4 −299.74∗ Monthly

We have also estimated a benchmark model, that is taken to bea simple homogeneous Poisson process alongwith an independentbinomial distribution for hurricane intensity. In this model thatignores the effects of ENSO, the mean rate of arrival is 0.6415hurricane per year and the probability parameter in the binomialdistribution is 0.3051. The standard errors of these estimations arerespectively 0.0778 and 0.0279.

As a first step, we compare models that have been estimatedwith annual data. Using the likelihood ratio test (LRT), we cancompare the benchmark model to model #1 since the former is aspecial case of the latter. With 4 additional parameters (degrees offreedom), one gets a p-value of 6.79%. Thus, considering the addedcomplexity,model #1has a significant better fit at a level of 10%butnot at 5%. As seen in the plot of Fig. 6, the failure to fit the increasedhurricane activity in the 1940s by model #1 may very well explainwhy the LRT has such a p-value. Model #1 could still be furtheranalyzed since (1) we cannot reject it at a level of 10%, (2) there isan interesting disparity between the parameters in each state and(3) this will have obvious impacts on the distribution of losses (seeSections 5.4 and 5.5).

We further continue our analysis of the fit of the models bychecking the appropriateness of a third state, i.e. by comparingmodels #1 and #2. The purpose of this test is to statisticallyverify the presence of another state in the ENSO phenomenon. Onenotices an important log-likelihood difference, i.e. 17.833. With 4additional parameters with respect to model #1, the LRT yields a


Parameters/function f Exponential Log–logistic Weibull Normal

a 1.6276 (0.4655) 1.5511 (0.4574) 1.3401 (0.5047) 1.2169 (0.3653)b 3.1691 (0.6159) 3.2208 (0.6362) 3.2863 (0.648) 3.3093 (0.516)αA 3.1930 (0.6157) 3.1931 (0.6162) 3.1928 (0.6051) 3.1929 (0.462)βA 2.3194 (0.4367) 2.3195 (0.4366) 2.3182 (0.429) 2.3177 (0.3195)αL 3.9603 (2.5588) 3.8142 (2.269) 3.2766 (1.6833) 3.0045 (0.5296)βL 3.6360 (2.1118) 3.4980 (1.9033) 2.9721 (1.4604) 2.7339 (0.5571)µ 6.8325 (1.334) 5.6490 (1.1665) 15.7881(15.8506) 5.5738 (1.2762)σ 0.5483 (0.3052) 5.8570 (2.2433)

Log-likelihood −299.79 −299.74 −299.63 −299.34



Table 9Expected hurricane losses in the next t years for all models given that the processstarts in La Niña.

t Model#1 #2 #4 Bench. #1 −Bench.Bench.

1 22326 21126 14046 13368 67%3 59130 58344 38299 40060 48%5 92424 90767 66504 66887 38%

10 174072 170641 161587 133719 30%

Table 10Standard deviation of hurricane losses in the next t years for all models given thatthe process starts in La Niña.


1 28924 28620 23849 22215 30%3 50189 50053 39463 38331 31%5 63667 64641 51756 49540 29%

10 89159 89937 81103 70046 27%

p-value of 3.4 × 10−7. Thus, a 3-state ENSO model is moreappropriate for the evolution of ocean temperatures.

Finally, we can compare models that use monthly data, i.e.models #3 and #4. One can see that accounting for parsimony (sayusing the Akaike or Bayes Information Criteria), model #3 has amuch better fit than model #4, further indicating the necessityof a Markov-switching framework to represent both hurricanefrequency and intensity.

5.4. Assessing individual losses

In this part of the numerical example, we apply the modelspresented above to evaluate the distribution of individual lossesfor a building located in a hurricane-prone area. We consider thebenchmark model along with models #1, #2 and #4. The reasonwhy we have excluded model #3 is because for long-term riskmanagement purposes, one does not need the exact timing ofevents within a year.

Suppose that the insured property is worth 100 000$ and is a1–3 story building having the characteristics of ‘‘Example-N’’ ofSection 5.1. The property is built somewhere in Florida such thatall hurricanes that occur in this state will cause damage. This willoverstate the riskmeasures and the results can bemademore real-istic by either using an attenuation function or use more localizedhurricane data (specific to a region instead of the whole state ofFlorida). We also assume that the state of ocean temperatures isknown as being in La Niña. This is to emphasize the importance ofENSO on the values of risk measures. The following results havebeen computed using 100 000 simulations and are presented inwhat follows.

Tables 9–12 show respectively the values of the expected losses,the standard deviation, the Value-at-Risk (at a level of 99%) andthe Tail VaR (also at a level of 99%) of X (t) for various modelsand values of t . The risk measures indicate that ignoring thedependence between hurricane frequency and intensity may havea significant impact on these quantities. The relative differencebetween the independent (benchmark) model and the proposedmodels is always as large as 25%–30%, and can be much larger inthe short run. One also notices that riskmeasures with the doubly-periodic model is somewhere in between the Markov-switchingmodels and the benchmark model. This is mainly because thelength of a cycle is stochastic in the proposed Markovian models,and fixed in the doubly-periodic model. This further increases thedamage potential during La Niña, rendering the doubly-periodicmodel less appropriate for long-term risk management purposes.

Table 1199% Value-at-Risk of hurricane losses in the next t years for all models given thatthe process starts in La Niña.


1 116458 117463 99022 93924 24%3 211329 208952 162473 157350 34%5 278929 279799 218640 210563 32%

10 424509 424037 383386 326745 30%

Table 1299% Tail Value-at-Risk of hurricane losses in the next t years for all models giventhat the process starts in La Niña.


1 136011 137763 118654 111488 22%3 242734 241772 187965 182358 33%5 317149 319042 249683 239307 33%

10 472607 472815 424557 363207 30%

5.5. Solvency of the insurance company

Suppose that a portfolio composed of 1000 insureds is exposedto a common hurricane risk. Moreover, the portfolio is such thateach property share the same insurance characteristics of thebuilding used in Section 5.4. Furthermore, we assume that the sizeand characteristics of the portfolio remain stable through time.

In this example, we analyze the ruin probability of an insurerthat ignores the dependence between hurricane frequency andintensity.We denote the surplus process of the insurance companyby U = {U (t) , t ≥ 0} where the surplus level at time t is U (t) =u + π t − S (t) and S (t) is as defined in (3). Based upon theresults of Section 5.4, we assume the premium income per unitof time π to be equal to 1.25 times the expected annual lossesin the independent model and the initial surplus to be twice aslarge as the expected annual losses in that model, meaning π =1.25 × E [SBENCH (1)] and U (0) = u = 2 × E [SBENCH (1)].

We denote by the rv τ the time of ruin where

τ =

infs>0

{s,U (s) < 0} , if U goes below 0 at least once

∞, 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 10 000 stochastic simulations to ap-proximateψ (u, t). Fig. 9 depicts the values of the ruin probabilityψ (u, t) under each model for t ∈ (0, 30], given that the state ofocean temperature is in La Niña.

For all three models, ignoring dependence between hurricanefrequency and intensity significantly aggravates the solvency of theinsurance portfolio, especially after approximately eight years. Forexample, the 15-year ruin probability is about 65% in model #1while it is about 40% in the benchmark model. Thus, the flow ofpremiums and the surplus should account for the state of ENSO toensure the solvency of the portfolio.

6. Concluding remarks

In the meteorological literature, it is documented that variousphenomena (like El Niño/Southern Oscillation (ENSO) and theAtlantic Multidecadal Oscillation) influence hurricane frequencyand intensity. In this paper, we have adapted models with aMarkovian environment to hurricane losses that account for thesemeteorological processes in a risk theory context. Moreover, wehave introduced frameworks for hurricane frequency, intensityand damage. The results that we obtain with Markovian modelsconfirm the existence of a low frequency/low intensity and a high



Fig. 9. Ruin probability of the insurer when premiums and surplus are based uponindependence between frequency and intensity of hurricanes.

frequency/high intensity state, that we have interpreted as beingEl Niño and La Niña respectively. This has an important effectin risk management as illustrated by the various risk measuresand solvency results. We found the doubly-periodic model of Luand Garrido (2005) to be less appropriate for risk managementpurposes because the duration of stay in La Niña is deterministic.

Onemay believe that hurricane losses seem to reach new recordhighs every year due to globalwarming. However, studies by Pielkeand Landsea (1998) and Pielke (2005) state that with normalizeddata (data corrected for inflation, changes in population andwealth), losses in the 1990s were lower than in any other decade.(Knutson et al. (2010) and IPCC (2013)) is that the total numberof storms in the Atlantic is likely to be stable or go down, butthe proportion of the strongest storms (category 3–5) is likely toincrease (i.e. fewer storms, but those that form will tend to bestronger). Hence, it would be interesting in the future to adapt theproposed models to take global warming into consideration.

Acknowledgments

All authors would like to acknowledge the financial sup-port from the Natural Sciences and Engineering Research Council(NSERC) of Canada and from the Chaire d’actuariat de l’UniversitéLaval. The authors would also like to thank Louis-Philippe Caron,researcher at the Catalan Institute of Climate Sciences (IC3)(Barcelona, Spain) and two anonymous referees for their com-ments on the paper.

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Author's personal copy - Université Laval · 2017. 9. 28. · Risk models with dependence between claim occurrences and severities for Atlantic hurricanes Mathieu Boudreaulta,, Hélène