PORTFOLIO THEORY FOR EARTHQUAKE INSURANCE RISK ASSESSMENT

PORTFOLIO THEORY FOR EARTHQUAKE INSURANCE RISK ASSESSMENT 63 Weimi DONG Ad Felix S WONG SUMMARY This paper presets a approach to quatifyig portfolio risks that ackowledges the importace of correlatio betwee losses at differet locatios (loss correlatio, i shorthad). The approach is evet-based: a group of evets ad their respective occurrece rates are idetified to represet the potetial overall risk i the regio, ad the loss of the portfolio for each evet as give by egieerig models is recorded. From these losses, the exceedace probability curve is developed to predict the probability that the portfolio loss will exceed a certai threshold. A key iovatio of the method is the itroductio of a geeral correlatio fuctio that embodies loss correlatio cotributios from geographic cocetratio, variability i buildig vulerability, ucertaity i soil amplificatio, ad choice of atteuatio model. These effects are quatified usig diversificatio factors which ca be computed readily o the policy ad portfolio levels. The variace of portfolio loss is the calculated based o this fuctio. The method is ot exact whe compared with the complete simulatio approach (which is curretly viable oly i theory), but eoys a attractive balace of accuracy ad computatio efficiecy that makes quatitative portfolio risk assessmet possible. Compatibility ad itegratio with cotemporary fiacial risk assessmet methodologies are discussed INTRODUCTION Isurace compaies that issue catastrophe policies such as for earthquakes ad hurricaes are cocered with their probable maximum loss (PML). Their portfolios cosist of may idividual policies, ad the maximum loss is a aggregatio of the losses from the idividual policies. Reisurace compaies that issue catastrophe treaties to primary isurers are also cocered with their portfolio risk. I this case, the risk is a aggregatio of the risks of the primary isurers portfolios. This paper discusses how portfolio losses ca be determied from their compoet (policy or cedat isurer) losses. For a portfolio with assets at may differet locatios, the mea of the aggregated loss is simply the sum of the mea locatio losses. However, the computatio of the stadard deviatio of the aggregated loss is more complicated because losses at ay two locatios may be correlated. Such correlatio is kow to have sigificat impact o the distributio of the aggregated loss. Less well kow but equally importat is the fact that the allocatio of losses to the reisurer, such as uder a excess-loss treaty, depeds ot oly o the mea of the aggregated loss but eve more so o the loss distributio. Hece, locatio loss correlatio plays a importat part i quatifyig portfolio risk. Loss correlatio ca materialize i two ways. The losses at ay two locatios due to all relevat evets (i.e., evets deemed to have a effect o at least oe of the locatios of iterest) may be correlated because the same evet(s) may lead to damage at both locatios. Such correlatio exists eve if the evet occurrece rates ad the correspodig loss estimates have o ucertaities. We refer to this type of correlatio, evets correlatio (ote the plural form for evet). The other type refers to correlatio of losses at two locatios give a evet, viz., how the ucertaity i the aggregated loss is affected by the ucertaities i the locatio losses due to a particular evet ad the tedecy of those ucertaities to vary together. We refer to this type as correlatio give a Risk Maagemet Solutios Ic, Melo Park, CA. USA. Email:weimid@ca.wai.com Weidliger Associates Ic, Los Altos, CA. USA. Email:felix@ca.wai.com

evet. Note that correlatio give a evet exists oly if the losses are ucertai. As we shall show, both types cotribute to the ucertaity i portfolio risk. The methodology that addresses the former is called the Evet-Loss Table (ELT) approach, while the method for the latter is called the global diversificatio factor approach. They will be described herei followig the presetatio of a simple example to highlight the importace of loss correlatio i isurace decisios, i case that is ot immediately obvious. We also idicate how the two sources of correlatio iteract with each other, ad how the ELT ad diversificatio factor methods ca be combied to form a complete methodology for risk assessmet that applies to ot oly isurace/reisurace portfolios but also fiacial ad market risks as well. EXAMPLE EFFECT OF CORRELATION ON INSURANCE LOSS ALLOCATION Cosider a portfolio with locatios ad the policy coverage is $, per locatio, for a total coverage of $ millio ( $,). The policy has a % deductible so that the amout of deductible is % of $ millio or $. millio as idicated i Fig.b. The treaty terms as show icludes FAC for Facultatives, QS for Quota Shares ad SS for Surplus Shares. Note that FAC ad FAC have % participatio, but differet attachmet poits ad limits. QS ad SS have participatio of % ad %, respectively, ad their attachmets ad limits as idicated i the figure. Suppose further that the mea damage to all locatios is the same, at % damage. Hece, the expected loss to the portfolio is $ millio (% of $ millio is $ millio) as idicated i Fig.a. For compariso, the distributio of the portfolio loss is preseted i Fig.c ad d for the totally idepedet ad totally correlated cases, respectively, assumig a coefficiet of variatio of.6 for each locatio loss. It is clear from these figures that i the totally idepedet case, the portfolio loss distributio is cocetrated (i.e., small stadard deviatio) ad the distributio is cetered o the determiistic value of $ millio (compare Fig.c with a). Whe the losses are totally correlated, the distributio is spread out over a wide rage (i.e., large stadard deviatio; see Fig.d). (a) o ucertaity (b) isurace treaty structure (c) ucertaities totally idepedet (d) ucertaities totally correlated Loss ($millio). FAC QS SS FAC. deductible % % % Figure. Isurace structure used i example. 63

These differeces i distributio have sigificat impact o Table. Compariso of Losses (i $) Allocated to Treaties of the Policy. the allocatio of No ucertaity Ucertaity, Ucertaity, isurer/reisurer Treaty but losses betwee but losses betwee losses, as a overlay of locatios are locatios are totally Fig.c (or Fig.d) with idepedet correlated Fig.b readily idicate. FAC,, 79,446 Numerical results FAC 8, summarized i Table QS,, 4,64 serve as a remider of SS 34,98 the importace of Reisurer Loss,, 33,6 icludig correlatio Isurer Loss 3, 3, 43,486 i portfolio loss Isured Loss,,,99 estimatio, although a Total Loss,,,,,, detailed discussio caot be icluded due to space limitatio. Note that Isured loss, isurer loss, ad reisurer loss are all effected due to correlatio give evet. THE EVENT LOSS TABLE APPROACH (EVENTS CORRELATION) Suppose that the risk for a property at a particular locatio is i the form of a group of evets (earthquakes from earby faults that are udged to have substatial effects o the assets should they occur) with various occurrece rates. Give that a evet has occurred, the loss sustaied by the property is computed usig stadard techiques. The evet losses for all relevat evets are the compiled ad collected i a table, called the Evet Loss Table or ELT (see Table ). Each row of the ELT correspods to a catastrophe evet i the group of credible scearios, ad is idetified by a umber, e.g., Evet ID =, with λ as the correspodig aual rate of occurrece. It is customary to arrage the evets accordig to the losses L. Note that for the momet λ ad L are assumed kow without ay ucertaity, i.e., they are poit estimate values. Ucertaities o rate ad loss will be brought i as a extesio of the ELT. Table. A Evet Loss Table (ELT). Evet ID Aual Rate Loss λ L λ L : : : λ L : : : λ L Suppose that each evet is a idepedet Poisso process. For the property of iterest, the the average aual loss E(L) ad the stadard deviatio of the loss σ are, respectively: E( L) = λ L () ad σ = = λ L = where the summatio idex correspods to the total umber of idepedet evets i the ELT, i.e., umber of rows i the table. Give the ELTs for two (properties at two) locatios, the correlatio betwee losses at the two locatios ca be readily established as follows. Let s deote the properties by A ad B. The ELTs for A ad B are rearraged i such a way that the evets i the tables are i the same order. I particular, L A, ad L B, are the respective losses from A ad B give evet with the evet rate λ. It is easy to see that for the combied loss, the average aual loss E(L) is simply, E( L) = λ( L, A + L, B) = (3) = E ( L) + E ( L) A B () 3 63

Furthermore, for property A, the stadard deviatio of loss accordig to Eq. is: σ = λ L, A A (4) whereas that for B, it is: σ = λ L, B B () O the other had, workig with the combied loss for both properties, the stadard deviatio of the combied loss is: σ = λ ( L, + L, ) AB A B (6) By defiitio, the, the correlatio coefficiet, ρ, ca be obtaied from Eqs.4-6 as: σ σ σ ρ = σ σ AB A B A B (7) Note that the cocept of ELT as show i Table is geeral, ad applies readily to a isurace compay s portfolio as well. I that case, etries i the loss colum deote, istead of Table 3. ELTs for a Idustry Portfolio ad Three Idividual losses for a particular property, the Compay Portfolios. portfolio losses for the evets. Evet Idustry Comp. A Comp. B Comp. C Aual Equatio 7 the defies the correlatio Loss Loss Loss Loss Rate betwee portfolios. As illustratio, 7. 6. cosider the (isurace) idustry s.. portfolio ad portfolios from three 3.. compaies as idicated i Table 3. Compay A has a % market share ad is totally correlated with the 4 6 8 6 4. 3.. 3.3.3.4 idustry loss. The correlatio 7 4..4 coefficiet, ρ A, is the. Compay B is partially correlated with the idustry ad, usig Eq.6 above, we calculate the correlatio coefficiet, ρ B, to be.77. Compay C is outside of the USA so that ay evet that affects Compay C does ot affect the U.S. idustry ad vise versa. Hece, the correlatio coefficiet for Compay C with U.S. idustry, ρ C, is. 8 9 3 4 6 3... 4 3 6 3 6 4...6.3..4.6.4. EXTENDED EVENT LOSS TABLE (EVENTS CORRELATION) Whe ucertaities associated with the rate of occurrece of a evet ad the loss estimate give a evet are take ito accout, the ELT approach ca be eriched with the respective ucertaity iformatio icorporated ito a exteded table, called exteded ELT or EELT for short. A example is show i Table 4, where the Rate ad Loss colums ow deote the mea rates ad mea losses, respectively, ad additioal colums 4 63

such as CV of Rate or CV of Loss deote the additioal iformatio o ucertaity. Eve higher levels of ucertaity iformatio ca be icluded i the same fashio. Table 4. A Exteded Evet Loss Table Showig Additioal Ucertaity Iformatio. Evet ID Mea Rate CV of Rate Mea Loss CV of Loss λ CV λ L CV L The developmet give i the previous sectio ca be readily applied to the EELT. I particular, it ca be show that with loss ad rate ucertaities icorporated, the stadard deviatio of a portfolio loss ca be approximated by { λ L CV } ( ) L = σ = + λ CV λ L CV L : : : : : λ CVλ L CV L : : : : : λ CVλ L CV L (8) which should be compared with Eq.. We ca use this formula to calculate the correlatio betwee two portfolios as doe with Eq.7 for the ELT. Like the ELT, the EELT as show i Table 4 is geeral, ad applies to a sigle locatio or a isurace compay s portfolio as well. I the latter case, etries i the loss colums deote, istead of losses for a particular property, the portfolio losses. Whereas the mea portfolio loss, L, is simply the sum of the compoet losses, the quatificatio of the CV of the portfolio loss ad, by extesio, the loss distributio, is by o meas trivial ad costitutes the mai subect of this paper. The reaso is correlatio, but this correlatio arises from ucertaities i the estimate of the compoet losses. It exists eve if we are cosiderig oly a sigle evet, as delieated i the followig. We call this correlatio give a evet. WEIGHTING FACTOR APPROACH (CORRELATION GIVEN AN EVENT) Now, we focus o a sigle evet (e.g., ay row i the EELT) ad the loss estimates for the portfolio i questio. Suppose the portfolio covers locatios (or cedat portfolios), ad deote the locatio losses by radom variables X i, i=, ad the portfolio loss by Y. Y is the a radom variable give by Y = Xi, i =, i (9) The task is: Give the mea loss for each locatio, X i, ad the stadard deviatio of loss for the locatio,σ Xi, compute portfolio mea, Y, ad the portfolio stadard deviatio, σ Y. As is well kow, the mea of the total loss for the portfolio, or portfolio mea, is: () Y = Xi, i =, i ad the variace of the total loss, or portfolio variace, is: Var[ Y] = σ = Var[ X ] + ρ σ σ, Y i i= i= = i+ () i Xi X ad ρ i, is the correlatio coefficiet betwee losses at locatios i ad. Note that ρ i, i Eq. is differet from ρ i Eq.7; ρ i, is the correlatio betwee two losses give a evet ad ρ i Eq.7 is the overall correlatio betwee two losses for all evets. While ρ i, ca be determied, i theory, by cosiderig all possible pairs of locatios accordig to Eq., such brute-force approach is impracticable. As a alterative, we suggest usig a global approach i the form of a weightig factor f such that the stadard deviatio of the total loss ca be approximated as follows: Ucertaities i occurrece rate ad property losses are discussed i a compaio paper i the same proceedigs (see [Wog, Che ad Dog., ]). Space limitatio excludes a presetatio of the derivatio of Eq.8, but the iterested reader ca cosult [Dog, 999]. 63

σ = f σ + ( f ) σ () Y Xi i= i= Xi The idea is to use, f, a udiciously chose weight for the portfolio, to reflect the portfolio s overall correlatio withi a scale defied by total correlatio (f = ) at oe ed ad total idepedece (f = ) at the other. The weight f is the a ecapsulatio of the maor factors affectig correlatio of losses at dispersed locatios. These factors are discussed below, ad a geeral approach to quatifyig them is outlied. Briefly, diversificatio factors are used to express the degree of diversificatio (cocetratio) i the policy/portfolio i these factors, i coformace to the fact that a well-diversified policy/portfolio has smaller loss correlatio whereas a cocetrated policy/portfolio has large loss correlatio. The diversificatio factors, i tur, are related to the equivalet, composite weight f. FACTORS INFLUENCING LOSS CORRELATION (DIVERSIFICATION) Maor factors of diversificatio (cocetratio) affectig policy/portfolio loss correlatio iclude geographic locatio, vulerability modelig, soil amplificatio ad hazard atteuatio modelig: Geographic Cocetratio: Observatios i past earthquakes poit out the existece of local pockets i which all structures suffered more (or less) severe damage as a group tha i eighborig areas. These pheomea are caused by local coditios such as basi effect, ad are ot icluded i geeral atteuatio models. From observatios of past evets, the swig due to these local effects ca cause a maximum of oe itesity level from the expected value. Assumig uiform distributio withi the same pocket, the itesity swigs for two locatios i the pocket will be totally correlated. The effect of geographic cocetratio o portfolio loss depeds o the portio of assets i such pockets. Parameter Ucertaity i Vulerability: If the mea damage ratio for a buildig class is actually larger tha the model, e.g., wood-frame buildigs i the Northridge earthquake, the the damage of all such buildigs will be uderestimated. Hece, the correlatio effect o portfolio loss is higher for portfolios cosistig of sigle buildig types tha for portfolios of may mixed types 3. From ATC data, the average coefficiet of variatio for the mea damage ratio is.387, which ca be used to calculate the correlatio coefficiet for portfolios cosistig of buildigs of the same class. If a portfolio is well diversified, say, uiformly distributed amog five classes, the the coefficiet of variatio will be smaller, amely,. 387 73. = ; there is much less correlatio. Ucertaity i Soil Amplificatio: Whe the buildigs i a portfolio are spread out over all types of soils, the correlatio i loss will be small compared that for a portfolio with buildigs o the same soil. For example, suppose soil amplificatio may lead to a swig i itesity of ±., uiformly distributed. The swig whe all buildigs are seated o the same soil will be correlated. Whe the buildigs are seated i four differet soils, the four uiformly distributed swigs will covolve ito a bell-shaped distributio. Atteuatio Model Ucertaity: If all buildigs are located at the same distace from the rupture, there is a good chace that all groud motio estimates are off due to the use of oe atteuatio model or aother, whe compared with portfolios with buildigs located at a wide rage of distaces from the rupture. I the latter case, some buildigs at a give distace may be uderestimated, while others at a differet distace may be overestimated. Let DG, Dc, Ds ad Dd represet the diversificatio factors for the above sources, respectively. Computer loss simulatio is used to determie the relatioship betwee these diversificatio factors ad the overall weightig factor f, which is stated below without elaboratio 4 : f =. 476. 37* D. 77* D. 38* D 997. * D (3) c s d G 3 It is assumed that parameter ucertaity i vulerability of oe buildig type is idepedet of that i aother type, i.e., vulerability of wood-frame buildigs may be uderestimated, but that of masory buildigs may be overestimated. 4 Details are ot preseted due to space limitatio but ca be foud i [Dog, 999]. 6 63

With f give by Eq.3, the stadard deviatio of the total loss show i the EELT ca be obtaied from Eq., ad the methodology is complete. SUMMARY AND FINANCIAL APPLICATIONS This paper has addressed correlatio issues that are crucial to portfolio loss estimatio. There is correlatio betwee the variatios of the mea losses of two locatios (or two portfolios) for differet evets; there is correlatio betwee ucertai losses of two locatios give a particular evet. Usig a overall weight factor that summarizes the correlatio cotributios from the mai physical elemets, the aggregated portfolio loss for differet participats (isurer or reisurer) ca be reasoably estimated. Igorig correlatio give the evet will sigificatly affect the loss allocatio amog the isured, isurer ad reisurer. The proposed EELT ad weight factor methodology ca be used for portfolio loss assessmet; it ca be used to estimate the average aual loss ad variatio of the loss for the portfolio. Furthermore, the method ca be used to quatify the covariace matrix of various regioal losses for a isurace portfolio, ad to support optimal capital allocatio (e.g., usig quadratic algorithms to miimize the overall variace [Markowitz, 99]). Similarly, a reisurace compay ca use the covariace matrix of various cedat portfolios to optimize its global exposure. I modelig fiacial portfolio risk,.p. Morga developed a applicatio called RiskMetrics to determie a quatity called VaR (Value at Risk, [Logerstaey ad Specer, 996]). It uses historical data to obtai the covariace matrix for differet sectors of the market, ad from this iformatio, derives the portfolio variace ad the percetile loss for the risk. I particular, the VaR of a portfolio of istrumets at 9% probability is give by: VaR = VRV T where V is a vector of VaR estimates for the respective istrumets, V = [ w 6. σ, w 6. σ, w 6. σ ] (4)..., () w i is the fractio of the ivestmet allocated to the i th istrumet, ad σ i is the stadard deviatio of the value of the i th istrumet (.6 σ i correspods to its 9% bouds i fluctuatio). R is the correlatio matrix, ρ.. ρ ρ.. ρ R =.......... ρ ρ.. ρ i is the correlatio betwee the values of the i th ad th istrumet, ad V T is the traspose of V. The RiskMetrics data bak provides all the istrumet performace statistics that are required by Eqs.4-6, otably, the stadard deviatiosσ i ad the correlatios ρ i used i Eqs. ad 6, respectively. Properties subect to catastrophe damage such as due to earthquakes ad hurricaes ca be viewed as aother form of asset or istrumet that is at risk, ad ca be treated by RiskMetrics i exactly the same way as foreig exchages or bods. I particular, the overall risk of a collectio of portfolios of properties is the aggregate of the costituet portfolio VaRs as computed by Eqs.4-6, whe the statistics of property damage due to catastrophes ad their correlatio are kow. Furthermore, real estate assets ca be assembled with other fiacial assets to form a balaced portfolio withi the RiskMetrics framework sice losses due to catastrophes are, uder most circumstaces, fairly idepedet of losses due to fiacial turmoil or currecy fluctuatios. The methodology described i this paper for catastrophe portfolio risk is compatible with the RiskMetrics approach. The portfolio loss correlatio coefficiets computed usig the EELT ad weightig fuctio ca be icorporated ito the covariace matrix i exactly the same way that risks of portfolios i foreig exchage ad bods are evaluated. (6) 7 63

Table below compares the requiremets of fiacial risk with catastrophe risk assessmet withi the framework of RiskMetrics. The colum uder catastrophe risk also serves as a road map for reviewig the developmets preseted i this paper. The ELT is a coveiet ad succict summary of the losses predicted by egieerig models. Startig from the basic portfolio ELT, the effects of ucertaities are icorporated ad summarized i a exteded portfolio table, the EELT, ad the portfolio EELTs are the combied per aggregatio eeds. Table. Fiacial Risk ad Catastrophe Risk Assessmet. Fiacial Risk Catastrophe Risk Obective Miimize risk, maximize retur; diversificatio; solvecy Database Vast historical database Very limited ad dated demographic chages valuatio chages idustry cocetratio chages applicability cocers Asset loss Fiacial models ad historical Egieerig models calibrated with limited data predictio data propagate ucertaities withi egieerig models ad approximatio aleatory ad epistemic ucertaities Correlatio betwee asset losses Historical data Diversificatio factors at the policy ad portfolio levels Portfolio Portfolio risk (theory) Aggregatig portfolios Portfolio risk (theory) Aggregate asset risks; use stadard deviatios ad correlatio coefficiets derived from data Aggregate portfolio risks; use stadard deviatios ad correlatio coefficiets derived from data Collectio of assets Aggregate asset risks; use stadard deviatios obtaied from egieerig model ucertaity aalysis, ad correlatio coefficiets derived from diversificatio factors Collectio of portfolios Aggregate portfolio risks; use stadard deviatios of portfolios, ad correlatio coefficiets derived from portfolio diversificatio factors ACKNOWLEDGEMENT The work preseted herei is partially supported by NSF grat CMS97-44, uder a oit proect betwee the Wharto School of the Uiversity of Pesyvalia ad RMS, Ic. REFERENCES. Dog, Weimi, Moder Portfolio Theory with Applicatio to Catastrophe Isurace, RMS Special Publicatio, 999.. Logerstaey,., ad Specer, M., RiskMetrics Techical Documet, Morga Guaraty Trust Compay ad Reuters Ltd., Fourth Editio, New York, December 996. 3. Markowitz, H. M., Portfolio Selectio Efficiet Diversificatio Ivestmet, Blackwell Publishers, 99. Wog, F., Che, H., Dog, W., ad, Ucertaity Modelig for Disaster Loss Estimatio, Proceedigs of WCEE, Aucklad, New Zealad, auary. 8 63