PRICING OF ZERO-COUPON AND COUPON CAT BONDS

Transcription

1 APPLICATIONES MATHEMATICAE 3,3 (23), pp Krzyszof Burnecki (Wrocław) Grzegorz Kukla (Wrocław) PRICING OF ZERO-COUPON AND COUPON CAT BONDS Absrac. We apply he resuls of Baryshnikov, Mayo and Taylor (998) o calculae non-arbirage prices of a zero-coupon and coupon CAT bond. Firs, we derive pricing formulae in he compound doubly sochasic Poisson model framework. Nex, we sudy -year caasrophe loss daa provided by Propery Claim Services and calibrae he pricing model. Finally, we illusrae he values of he CAT bonds ied o he loss daa.. Inroducion. Caasrophe bonds, also called CAT bonds, are insurance-linked securiies ha enable insurers o ransfer he risk of naural disasers like earhquakes or hurricanes o capial markes. They are sold o large insiuions. A financial inermediary, a reinsurance company or an invesmen bank, issues a bond o a paricular insurable even like e.g. Los Angeles earhquake. The proceeds from he sale are pu ino a collaeral accoun. If here is no earhquake, invesors are paid generous ineres rae, bu if he earhquake occurs and he claims exceed a specified amoun, invesors sacrifice fully or parially heir principal and ineres. For insurers he deals creae a pool of money ha can be apped immediaely ino a disaser. CAT bonds appeal o professional money managers because caasrophe risk is a new asse class ha is uncorrelaed wih socks and radiional bonds. They are growing in imporance also because insurance capaciies worldwide have been severely reduced by he evens of Sepember 2. In he aricle of Baryshnikov e al. (998) he auhors presened an arbirage-free soluion o he pricing of CAT bonds under condiions of coninuous rading. They modelled he sochasic process underlying he CAT bond as a compound doubly sochasic Poisson process. In Secion 2 2 Mahemaics Subjec Classificaion: 62P5, 62-7, 6G55, 6H3. Key words and phrases: caasrophe bond, doubly sochasic Poisson process, loss disribuion, non-arbirage price, non-parameric es. [35]

2 36 K. Burnecki and G. Kukla we apply heir resuls o calculae non-arbirage prices of a zero-coupon and coupon CAT bond. In Secion 3 we sudy -year caasrophe loss daa provided by Propery Claim Services (PCS). We find a disribuion funcion which fis he observed claims in a saisfacory manner and esimae he inensiy of he Poisson process governing he flow of he naural evens. In Secion 4 we illusrae he values of he CAT bonds associaed wih he loss daa wih respec o he hreshold level and mauriy ime. To his end we apply Mone Carlo simulaions. 2. Compound doubly sochasic Poisson pricing model. The CAT bond price process V, [, T ], is modelled by many facors: ype of region, kind of loss even, sor of insured propery ec. Baryshnikov e al. (998) describe he bond by means of he aggregae loss process L and he rigger value D. Se a probabiliy space (Ω, F, F, ν) and an increasing filraion F F, [, T ]. The CAT bond is well characerized by: A doubly sochasic Poisson process (see Bremaud 98) M s (s [, T ]) describing he flow of naural evens of a given ype in he region. The inensiy of his process is assumed o be a predicable bounded process m s. The losses {X i } i N, which are independen and idenically disribued wih F (x) = P{X i < x}. Moreover, X and M are independen. The aggregae loss process L = M i= X i. A pogressive process of discouning raes r. We assume he process is coninuous a.e. This process describes he value a ime s of USD paid a ime > s by ( exp( R(s, )) = exp s ) r(ξ) dξ. The mauriy ime T and hreshold level D. The hreshold ime τ = inf{ : L D}, which is he momen when he aggregaed loss L exceeds he value D. In he case of a zero-coupon bond: paymen of a cerain (random) amoun Z a mauriy ime T coningen on hreshold ime τ > T. In he case of a coupon bond: paymen of he principal (face value) a mauriy ime T coningen on hreshold ime τ > T and coupon paymens C which sop immediaely a τ (in general a process which is predicable and coninuous on [, T ]). Now define a new process N = {L D}. Baryshnikov e al. (998) in Proposiion 2 show ha his is also a doubly sochasic Poisson process wih he inensiy () λ s = m s ( F (D L s )) {Ls <D}.

3 Pricing of zero-coupon and coupon ca bonds 37 Le us consider a bond wih he paymen of a cerain amoun Z a mauriy ime T coningen on hreshold ime τ > T, which is in fac a zero-coupon CAT bond. Define he process Z s = E(Z F s ). The condiion required is ha Z s is a predicable process, which can be inerpreed o mean ha he paymen a mauriy is no direcly linked o he occurrence and iming of he hreshold. The amoun Z can be e.g. he principal plus ineres which is usually defined as a fixed percenage over LIBOR. We obain he following resul. Theorem 2.. The non-arbirage price of he zero-coupon CAT bond associaed wih he hreshold D, he caasrophic flow M s, and he disribuion funcion F of he incurred losses, paying Z a mauriy, is given by ( [ T V = E Z e R(,T ) ] ) m s [ F (D L s )] {Ls <D} ds F. Proof. Clearly he price of a CAT bond paying Z a mauriy a ime < τ is V = E(Ze R(,T ) ( N T ) F ). Represening N T as T dn s one arrives a he expression ( V = E Ze R(,T )( T dn s ) F ). From he definiion of a doubly sochasic Poisson process (see Bremaud, 98) we have ( V = E Ze R(,T )( T ) ) λ s ds F. Now we apply () o ge he asserion. We noe ha Baryshnikov e al. (998) in p. 3.2 of heir aricle presened a price of he hreshold bond paying Z a mauriy, which corresponds o he case of he zero-coupon CAT bond, bu heir formula is incorrec. Now, we consider a CAT bond wih only coupon paymens C which sop immediaely a τ. The following heorem gives he fair CAT bond price formula for such a bond. Theorem 2.2. The non-arbirage price of he CAT bond associaed wih he hreshold D, he caasrophic flow M s, and he disribuion funcion F of he incurred losses, wih coupon paymens C s which cease a hreshold ime τ, is given by ( T V = E s e R(,s) C s [ ] m ξ [ F (D L ξ )] {Lξ <D} dξ ds F ).

4 38 K. Burnecki and G. Kukla Proof. I is easy o see ha he price of a CAT bond wih coupon paymens C o he hreshold even τ is (T V = E e R(,s) (2) C s ( N s ) ds F ). Repeaing he seps as in he proof of Theorem 2. we obain he conclussion. We noe ha Baryshnikov e al. (998) (p. 3.2) sudied he case of coupon paymens P s on he hreshold bond, which corresponds o he above case. In he proof of Theorem 2.2 we jus apply heir saring formula which can be rewrien as (2). We also remark ha, unforunaely, hey furher use inegraion by pars incorrecly. Finally, we combine Theorems 2.2 and 2. wih Z = face value (FV) in order o obain he following fair price for he coupon CAT bond. Corollary 2.. The non-arbirage price of he coupon CAT bond associaed wih he hreshold D, he caasrophic flow M s, he disribuion funcion F of he incurred losses, paying FV a mauriy, and coupon paymens C s which cease a hreshold ime τ, is given by ( T V = E FV e R(,T ) + e R(,s)[ s C s ( ] FV e R(s,T ) m s [ F (D L s )] {Ls <D} ds F ). ) m ξ [ F (D L ξ )] {Lξ <D} dξ In he foregoing cases (see Theorem 2. and Corollary 2.) we assumed ha in he case of he rigger even he principal is fully los. However, if we would like o incorporae a parial loss in he conrac i is jus enough o muliply Z by a proper consan. 3. Calibraion of he pricing model. We conduced empirical sudies for daa obained from Propery Claim Services. The daa (see Figure ) concern he US marke s loss amouns in USD, which occurred beween 99 and 999 and were adjused using he discoun window borrowing rae (he discoun rae refers o he simple ineres rae a which deposiory insiuions borrow from he Federal Reserve Bank of New York). Only naural perils like hurricane, ropical sorm, wind, flooding, hail, ornado, snow, freezing, fire, ice and earhquake which caused damages exceeding USD 5 m were aken ino consideraion. We noice ha peaks in Figure mark he occurrence of Hurricane Andrew (24 Augus 992) and Norhridge Earhquake (7 January 994). In order o calibrae he pricing model we have o fi boh he disribuion funcion of he incurred losses F and he process M governing he flow of

5 Pricing of zero-coupon and coupon ca bonds 39 naural evens. We commence wih he presenaion of ypical claim size disribuions. 2 x.8.6 Adjused PCS caasrophe claims (USD) Years Fig.. Illusraion of he PCS caasrophe loss daa, Claim amoun disribuions. The claim disribuions, especially describing propery losses, are usually heavy-ailed. In he acuarial lieraure, o describe such claims coninuous disribuions are ofen proposed (wih he domain R + ): lognormal disribuion, wih he disribuion funcion (d.f.) given by ( ) x ln x µ F (x) = Φ = e ln y µ ( ) 2 σ 2 dy, x >, σ >, µ R, σ 2π σy where Φ(x) is he sandard normal (wih mean and variance ) d.f.; Pareo disribuion, wih he d.f. ( ) λ α F (x) =, x >, α >, λ > ; λ + x Burr disribuion, wih he d.f. ( ) λ α F (x) = λ + x τ, x >, α >, λ >, τ > ;

6 32 K. Burnecki and G. Kukla gamma disribuion, wih he d.f. F (x) = x Γ (α)β α yα e y/β dy, x >, α >, β >. The choice of he disribuion is very imporan because i influences he bond price Non-parameric ess. The derivaion of claim size disribuions from he loss daa could be considered o be a separae discipline, which requires applying mehods of mahemaical saisics (cf. Daykin e al. 994). The objecive is o find a disribuion funcion F which fis he observed daa in a saisfacory manner. The approach mos frequenly adoped in he acuarial lieraure is o find a suiable analyic expression which fis he observed daa well and which is easy o handle (see e.g. Burnecki e al. 2). Once he disribuion is seleced, we mus obain parameer esimaes. In wha follows we use he momen and maximum likelihood esimaion. The nex sep is o es wheher he fi is adequae. This is usually done by comparing he fied and empirical disribuion funcions, more precisely, by checking wheher values of he fied disribuion funcion a sample poins form a uniform disribuion (cf. D Agosino and Sephens 986). To his end we apply he well known and no so well known non-parameric ess, namely χ 2, Kolmogorov Smirnov (KS), Cramer von Mises (CM) and Anderson Darling (AD), verifying he hypohesis of uniformiy (see e.g. Kukla 2). A very naural and well known es is he χ 2 saisic k χ 2 k = k (n i n/k) 2, n i= where n is he overall number of observaions and n i is he number of observaions which fall ino he inerval [(i )/k, i/k]. χ 2 k has an approximae chi-square disribuion wih k degrees of freedom. In general, he beer he fi, he smaller χ 2 k. This es becomes more discriminaing as he sample size becomes larger. Anoher classical measure of fi is he Kolmogorov Smirnov saisic D n = sup F (x) F (x), x R where he empirical d.f. is defined as F (x) = n n {xi x}. i= The saisic D n measures he disance beween he empirical and fied d.f. in he supremum norm.

7 Pricing of zero-coupon and coupon ca bonds 32 The oher wo ess we apply are he Cramer von Mises and Anderson Darling ess. The former uses he saisic C n = n ( F (x) F (x)) 2 df (x) while he laer (considered o be he bes wihin he class of ess based on empirical d.f.) uses AD = n ( F (x) F (x)) 2 df (x). F (x)( F (x)) In order o inerpre he resuls of he ess we compare hem wih he corresponding criical values C α (for he same significance level α). When he value of he es is less han he corresponding C α we accep he fi as adequae (here is no reason o rejec he null hypohesis). The criical values C α of he ess given a significance level α (e.g. α =.5) can be found in he lieraure (see e.g. D Agosino and Sephens 986, and Sephens 974) Resuls of he fi procedure. Firs we sudied he loss amouns. Disribuions were fied using he momen and maximum likelihood esimaion. The resuls of he parameer esimaion and es saisics are presened in Table. The lognormal disribuion wih parameers µ = and σ =.348 passed all ess (he corresponding es saisics are in boldface), so we chose i for furher analysis. Table. Parameer esimaes and es saisics for he caasrophe loss amouns Disribuions: lognormal Pareo Burr gamma Parameers: µ=8.446 α= α=3.883 α=.9796 σ=.348 λ=3.32e+8 λ=.89e+5 β=.6348e+8 Tes values (in brackes criical values for α =.5): τ=.547 χ 2 (3.44) D n (.729) C n (.468) AD (2.492) Nex, we fied he doubly sochasic Poisson process governing he occurrence imes of he losses. We sared he analysis wih he simples case assuming he inensiy m s is equal o a non-negaive consan m. Sudies of he quarerly numbers of losses and he ineroccurrence imes of he caasrophes led o he conclusion ha he flow of he evens may be described by he Poisson process wih he daily inensiy m =.95.

8 322 K. Burnecki and G. Kukla The parameers of he fied model imply ha he expeced value of a daily loss is USD me µ+σ2 /2 USD 9 m. 4. Dynamics of he CAT bond price. In his secion we presen CAT bond prices for = days, namely V. To his end we apply he appropriae formulae and Mone Carlo simulaions. We assume for illusraion purposes ha he coninuous discoun rae r equivalen o LIBOR = 2.5% is consan and equal o ln(.25), T [9, 72] days, D USD [.7, 8.55] bn (quarerly o 5 quarerly average loss) and he principal is USD. Furhermore, in he case of he zero-coupon CAT bond we assume ha he bond is priced a 3.5% over LIBOR. If here is no rigger even, he oal yield is hence 6% and we pu Z = USD.6. Figure 2 depics he zerocoupon CAT bond values wih respec o he hreshold level and expiraion ime (cf. Theorem 2.). We can see ha he greaer he expiraion ime, he lower he CAT bond value, and increasing he hreshold level leads o greaer prices. When T = 9 days and D = USD 8.55 bn he CAT bond price approaches he value USD.6e ln(.25)/4 USD.53, which corresponds o he siuaion when τ T w.p.. Value of he bond (USD) x Threshold level (USD) Expiraion ime (days) 7 8 Fig. 2. The zero-coupon CAT bond price wih respec o he hreshold level and expiraion ime Before we presen he case of he coupon bond, we concenrae on he bond paying only coupons (cf. Theorem 2.2). We assume C.6. The values of V are depiced in Figure 3. We clearly see ha now he siuaion is quie differen, namely he price increases when he expiraion ime and

9 Pricing of zero-coupon and coupon ca bonds 323 hreshold level are greaer. When D = USD 8.55 bn and T = 72 days he 72 price of he bond approaches USD.6 e ln(.25)/36 d USD.8, which corresponds o he siuaion when τ T w.p...8 Value of he bond (USD) x Threshold level (USD) Expiraion ime (days) 7 8 Fig. 3. The CAT bond price, for he bond paying only coupons, wih respec o he hreshold level and expiraion ime Value of he bond (USD) x Threshold level (USD) Expiraion ime (days) 7 8 Fig. 4. The coupon CAT bond price wih respec o he hreshold level and expiraion ime

10 324 K. Burnecki and G. Kukla Finally, we consider he case of he coupon CAT bond. We assume as previously ha C.6. We can see he CAT bond prices in Figure 4. The influence of he hreshold level D on he bond value is clear as in he cases discussed above bu he effec of changing he expiraion ime T is no sraighforward. As T ges bigger he chance of receiving more coupons is greaer bu a he same ime he possibiliy of loosing he principal of he bond increases. In our case (see Figure 4) he price decreases wih respec o he expiraion ime bu his does no have o be always rue. We also noice ha he bond prices in Figure 4 are lower han he corresponding ones in Figure 2, bu we mus remember ha in he laer case Z = USD.6 and now we have Z = USD (cf. Corollary 2.). Acknowledgemens. The firs auhor hankfully acknowledges he suppor of he Sae Commiee for Scienific Research (KBN) Gran No. PBZ-KBN 6/P3/99. References R. B. D Agosino and M. A. Sephens (986), Goodness-of-Fi Techniques, Marcel Dekker, New York. Yu. Baryshnikov, A. Mayo and D. R. Taylor (998), Pricing of CAT bonds, working paper (hp:// P. Bremaud (98), Poin Processes and Queues. Maringale Dynamics, Springer, New York. K. Burnecki, G. Kukla and R. Weron (2), Propery insurance loss disribuions, Physica A 287, C. D. Daykin, T. Penikainen and M. Pesonen (994), Pracical Risk Theory for Acuaries, Chapman & Hall, London. G. Kukla (2), Insurance Risk Derivaives, MSc hesis, Wrocław Univ. of Technology. M. A. Sephens (974), EDF saisics for goodness-of-fi and some comparisons, J. Amer. Sais. Assoc. 69, Hugo Seinhaus Cener for Sochasic Mehods Insiue of Mahemaics Wrocław Universiy of Technology Wybrzeże Wyspiańskiego Wrocław, Poland burnecki@im.pwr.wroc.pl kukla@im.pwr.wroc.pl Received on (653)