THE USE OF RISK ADJUSTED CAPITAL TO SUPPORT BUSINESS DECISION-MAKING

Transcription

1 THE USE OF RISK ADJUSTED CAPITAL TO SUPPORT BUSINESS DECISION-MAKING By Gary Patrk Stefan Bernegger Marcel Beat Rüegg Swss Rensurance Company Casualty Actuaral Socety and Casualty Actuares n Rensurance 1999 Call for Papers CARe meetng, Baltmore, Md. June 6-8, 1999

2 OUTLINE: 0. Abstract 1. Pressures on Captal (Surplus) 2. How Much Captal Do You Really Have? Rsk Bearng Captal (RBC) 3. How Much Captal Do You Really Need? Rsk Adusted Captal (RAC) 4. How To Estmate RAC: Underwrtng Model 5. Modelng Man-made Maor Catastrophes (Threats) 6. How To Estmate RAC: Investment Model 7. How To Estmate RAC: Credt Model 8. How To Estmate RAC: Puttng It Together 9. How to Allocate RAC to Lne, Product, etc. 10. Managng RAC To Optmze Rsk and Return 11. Concluson 12. Bblography 13. Appendces CARepaper\paperFN9.doc 2

3 0. Abstract Ths s partally a conceptual paper about the reasons why an nsurance company should address rsk and captal ssues n a methodcal manner and about the problems encountered dong so. But t also offers some mathematcal methods for dealng wth some of the problems. We do not offer the reader the fnal answer, snce we certanly don t have t. But we do offer some deas and some procedures for obtanng useful measurements. Wthout reasonably accurate parameter estmaton, the most sophstcated dynamc fnancal analyss model s smply a black box mappng nformaton accordng to the garbage n, gospel out syndrome (let us all bow down to our computers and worshp ther unarguable output!). Modelers of nsurance rsk may fnd value n the dscusson of modelng man-made maor catastrophes va the constructon of threat scenaros. The secton on modelng nvestment rsk dscusses possble ways of usng the prevalng alue at Rsk model and some problems n dong so. The secton on credt rsk outlnes the modelng problems encountered here. The reader may fnd the dscusson of captal allocaton to be partcularly enlghtenng. In the secton on managng rsk adusted captal (RAC), we attempt to show, as smply as possble, how the concept of RAC can be used by management to steer the course of busness decson-makng. The Bblography lsts some references whch the reader can use to learn more about the deas presented n ths paper. And the Appendces contan more mathematcs about some of the models and ther estmaton. At ths stage we also want to menton that the overall captal estmaton and allocaton CARepaper\paperFN9.doc 3

4 methodology descrbed n ths paper s ntended for a company s nternal rsk management. It cannot be used n the same way by external partes such as regulators, ratng agences, etc. These external partes need a standard model for the whole ndustry, and they must rely only upon publcly avalable nformaton. 1. Pressures on Captal (Surplus) We use the terms captal and surplus nterchangeably throughout ths paper. The pressures on nsurance ndustry captal are ntense and conflctng. Company shareholders, polcyholders, nsurance regulators and ratng agences are all pushng and pullng n dfferent drectons. The shareholders want ther captal to perform, that s, earn a hgher return. But there are many obstacles. Industry returns-onequty (RoE) have been weak hstorcally; most crtcs see them as beng less than commensurate wth the rsk level. In addton, long-latent clams are stll a drag upon the results of many companes. But yet, many people beleve there s excess capacty currently n the nsurance ndustry. Rates are decreasng, thus drvng down proft margns. It s a stuaton of too much captal chasng too lttle busness. To satsfy shareholders hopng to obtan a hgher return, there s an ntense compettve push to assume more rsk n order to use captal more effcently. Meanwhle, polcyholders, nsurance regulators and ratng agences are all pullng n the drecton of hgher captalzaton. They are concerned about nsurance company solvency n lght of the recent greater recognton of the ndustry s extreme exposure to natural catastrophes, the emergence of clams stemmng from many long-latent CARepaper\paperFN9.doc 4

5 man-made exposures and the threats of future clams from many smlar exposures. The recent savngs and loan crss n the US has made nsurance regulators and ratng agences aware that such a crss could also possbly occur n the nsurance ndustry f a clams shock s accompaned by a fnancal shock. In order to pull nsurance companes to a hgher, more conservatve captal base, the NAIC has formulated the concept of rsk-based captal to defne relatvely hgh captal thresholds for companes operatng n the US [ref. 1.1]. Wth these ntense, conflctng pressures, nsurers need a better concept of captal n order to measure captal adequacy and to help steer decson-makng throughout ther companes. We wll dscuss varous knds of captal. The three man types we dstngush are: (1.1) Types of captal Publcly-perceved captal: Ths has more than one value. These are the varous values of captal calculated by the statutory or GAAP fnancal statements, the NAIC, A.M. Best, Standard & Poors, etc. These are external vews. Rsk bearng captal (RBC): In secton 2, we wll defne a smple calculaton of the captal that the company has avalable to support ts busness. Note that the RBC gves an nternal vew, and s dstnctly dfferent from the NAIC rsk based captal concept, whch we would classfy as one of the publcly-perceved types of captal. CARepaper\paperFN9.doc 5

6 Rsk adusted captal (RAC): In secton 3, we wll defne a smple calculaton of the captal that the company needs to support ts busness. Agan, ths wll be an nternal vew. 2. How Much Captal Do You Really Have? Rsk Bearng Captal (RBC) The smplest answer to the queston of how much captal you really have s fnancal statement captal, ether statutory or GAAP. Ths of course equals fnancal statement assets mnus fnancal statement labltes. It has the advantage of beng very smple. It also has the advantage of beng audted; t s ndependently verfed and sgned-off by professonals who are potentally lable for neglgence f, for example, the future run-off of loss reserves turns out sgnfcantly dfferent from that stated. It s also publc nformaton, prnted n black and whte, for revew and comment by any crtcs or other nterested partes. A problem wth fnancal statement captal s that t doesn t gve a complete pcture of the value of an nsurance company. The tme value of money s generally not recognzed for property and casualty companes. It does not recognze varous hdden values such as goodwll. But worst of all, t s a snapshot pcture. It s not a dynamc vew of an ongong, actve company. It s not forward lookng. It looks backward only to prevous exposure. CARepaper\paperFN9.doc 6

7 A better vew of how much captal a company really has to support the rsk generated by ts busness s gven by the concept of rsk bearng captal, or RBC. A very smple, operatonal vew of RBC can be obtaned as follows. (2.1) RBC = fnancal statement captal plus any unrealzed captal gans not ncluded above plus the dscount nherent n the loss reserves plus other hdden values mnus latent taxes The latent taxes are those that would occur f the three plusses lsted above flowed through ncome. One can argue ad nauseam about whch fnancal statement to start wth: statutory or GAAP. Clearly, whole chapters of lengthy fnancal accountng books can be wrtten about the exact treatment of the other tems. And actuares can go on for days about how to dscount loss reserves, at what nterest rates, etc. We do not wsh to prescrbe too much here. Snce we ntend ths to be an nternal vew, we beleve t s up to the ndvdual companes and ther techncal staffs to decde how exact a measurement they want. The man thng s to do somethng along these lnes. Don t worry too much about dottng s and crossng t s. Snce there are so many fuzzy ssues and dffcult-to-measure varables n any endeavor lke ths, tryng to be overly exact s wasted effort. CARepaper\paperFN9.doc 7

8 The man pont s to devse for your company some measure of rsk bearng captal n order to gve management a reasonably accurate pcture of the amount of captal they have avalable to support current and possble future busness. 3. How Much Captal Do You Really Need? Rsk Adusted Captal (RAC) Perhaps the queston that should be asked frst s: Why must we have any captal at all? If we were dealng wth a stuaton where run-off of current labltes and the earnngs on current assets were completely predctable and where the company was not about to assume addtonal labltes and assets n the comng year, there would be no need for captal. The concept of captal makes no sense wthout the concept of rsk. And rsk has entrely to do wth the unpredctablty of future events. The clams run-off wll never be the same as predcted; the future earnngs on and the future values of current assets wll be, except n unusual crcumstances, also unpredctable. And of course future assumed rsk-transfer busness by defnton s unpredctable. Captal s necessary for the future. It s not a statc concept for ether clams run-off or for ongong busness. Ths tells us why NAIC rsk-based captal or most ratng agency models do not gve us good answers to the queston of how much captal do you really need. These are relatvely smple models; they are quck n drty, geared to adequacy, not CARepaper\paperFN9.doc 8

9 optmzaton. They usually have faulty thermostats, employng smple ratos of premum or loss reserves whch ndcate less requred captal when rates or reserves are nadequate, whch s of course exactly when you need more captal. They also don t recognze management or shareholder rsk-level preferences. Lke fnancal statement captal, they are retrospectve, not prospectve. Fnally, they are not easly translatable to lnes of busness, types of contracts, proft centers, etc., n a manner that s useful for supportng busness decson-makng. We may classfy the rsks we should consder when defnng rsk adusted captal for an nsurance company. (3.1) Insurance company techncal rsks Underwrtng rsk Clams Ratng system bases: parameters, formulas, etc. Some actuares would further splt ths accordng to the concepts of parameter rsk versus process rsk, or splt even further wth the concept of model specfcaton rsk. Underwrtng cycles Investment rsk Market rsk: stock market, nterest rate, foregn exchange Default CARepaper\paperFN9.doc 9

10 Lqudty Etc. Credt rsk Rensurance Accountng balances due Letters of credt Etc. Note that we consder only techncal rsks, gnorng softer concepts whch mght be gathered under the rubrc management rsk. These other knds of rsks are better handled outsde of a techncal model. What crtera do we want our defnton of rsk adusted captal to satsfy? We want RAC to be the level of captal an nsurance company needs to wrte ts busness. Among many possble crtera, we select the followng. (3.2) Crtera for RAC It must meet specfed management rsk and survval crtera. It should quantfy the rsk/return trade-off for all rsk exposures. It must be useful for makng approprate rsk-based busness decsons. Management rsk crtera have to do wth the publc statements of the company s results. These crtera may have to do wth year by year fluctuaton of results or wth downsde potental. Although fluctuatons are annoyng, the real fear of management CARepaper\paperFN9.doc 10

11 s downsde potental. They fear that an event or a seres of events may occur that mght cause the company s publcly-perceved captal to fall far enough to nterfere wth the company s ablty to contnue normal busness and also to rase serous questons about the company s contnuaton. Snce t s absolutely crtcal that the model reflects management rsk tolerance, t s absolutely crtcal for management to understand enough of the model so that they can understand what s beng asked of them, so that ther opnons are translated accurately nto the model structure and parameterzaton. Thus we propose a very smple, non-black box model for RAC. There are three steps n constructng our very smple RAC model. (3.3) A very smple RAC model Management specfes a smple rsk tolerance rule. The probablty dstrbuton of the company s result s estmated for the tme perod specfed by the management rsk tolerance rule. RAC s set to be the mnmum RBC necessary at the begnnng of the tme perod so that the RBC at the end of the tme perod satsfes the management rsk tolerance rule. As an example, n ths paper we wll use a very smple management rsk tolerance rule. (3.4) A very smple management rsk tolerance rule CARepaper\paperFN9.doc 11

12 Defne the management Rsk Tolerance Level (RTL) to be that value of RBC necessary to mantan a gven external ratng, e.g., A.M. Best A ratng, S&P BBB ratng, etc. Wth a specfed probablty, e.g., 90%, 95% or 99% At the end of a tme perod of one, two or n year(s). You can see that ths s ndeed a very smple management rsk tolerance rule. To smplfy our dscusson, let us assume that the probablty s 99% and the tme perod s one year. A tme perod of one year s not long enough to completely model the effects of potental shocks to captal caused by the manfestaton of long-latent clams. And of course a sngle probablty level s also very smplstc: why not use 99.5%? 98%? And of course management has more on ther mnd than the company s A.M. Best ratng. However, let us walk wth management before we make them (and ourselves) run. Ths defnton of RTL s the same as the defnton of u C, the early warnng lmt of captal n Daykn, Pentakänen and Pesonen s Practcal Rsk Theory for Actuares [ref. 3.1, p.365]. Note that the RTL s defned to be a mnmal level of RBC, but yet the frst crteron refers to a company s selected external ratng that utlzes one of the publcly-perceved types of captal. Thus we must construct a mappng of the RTL level of RBC onto the selected external ratng s captal level. Rather than gettng bogged down n ths constructon, we leave ths detal to the reader (remember not to worry too much about s and t s.). CARepaper\paperFN9.doc 12

13 By the company s result durng the tme perod (n our case, one year), we mean smply the change n RBC from begnnng to end. (3.5) Company s Result CR = { endng RBC} mnus { begnnng RBC} Note that CR s a calendar year concept. CR may be thought of, and modeled, as the sum of three components. (3.6) CR = company s calendar year underwrtng result plus plus company s calendar year nvestment result company s calendar year credt result A smple graph of the probablty densty functon of CR s llustrated n (3.7). CARepaper\paperFN9.doc 13

14 (3.7) CR probablty densty functon 1% $0 Expectaton $Result Note that the dstrbuton, whose random varable s essentally calendar year premum plus nvestment result mnus expenses mnus ncurred clams, s decdedly non-normal. It s skewed to the left snce the dstrbuton of ncurred clams s skewed to the rght [ref ] and the dstrbuton of the nvestment result s usually modeled to be essentally normal [ref. 3.4]. What else can we say about the dstrbuton of CR? The expectaton s the expected (planned?) result for the calendar year. The dstrbuton obvously depends upon the degree of dversfcaton n the company s underwrtng, nvestment and credt rsk portfolos: the more dversfed (lower correlaton), the narrower the spread. We can now construct a smple pcture of how RAC s defned. CARepaper\paperFN9.doc 14

15 (3.8) Defnton of RAC RBC Expected RBC RAC RTL Tme 1% (year) 0 1 year Gven ths smple model and defnton of RAC, we have the followng relatonshps. (3.9) RTL = RAC + CR 1% or RAC = RTL - CR 1% where CR 1% s the frst percentle of the dstrbuton of CR. In order to test our understandng of RAC (may we say RACology?), we can ponder the followng results. (3.10) Some smple RAC results As dversfcaton ncreases (lower correlaton among rsk portfolo segments), RAC decreases. Thus the amount of RAC relatve to rsk CARepaper\paperFN9.doc 15

16 volume, e.g., premum, loss reserves, assets, etc., provdes one measure of dversfcaton. As RTL decreases, RAC decreases. As the probablty (99%) decreases, RAC decreases. As the tme perod decreases, RAC decreases. As RAC decreases, the company s RBC can be reduced or addtonal busness can be wrtten. As a consequence, the company s needed target prcng margns wll decrease. 4. How To Estmate RAC: Underwrtng Model Our smple RAC depends upon the dstrbuton of CR, the company s calendar year result, and CR has three components: underwrtng, nvestment and credt results. The frst step s to estmate the dstrbuton of the company s calendar year underwrtng result. We defne the underwrtng result, CR U, as follows: (4.1) CR U = calendar year net earned premums mnus calendar year net expenses mnus calendar year net ncurred losses all dscounted at selected rsk-free nvestment returns The net result should however not be modeled drectly, but va a model for the gross result and a separate rsk transfer model for ceded rensurance. Ths approach then CARepaper\paperFN9.doc 16

17 allows us to address separately n the credt result model the mpact of the credt rsk arsng from the ceded rensurance. In the followng we wll descrbe the modelng of the gross result. Why complcate the underwrtng model by reflectng rsk-free nvestment ncome? Why not smply leave all the nvestment ncome n the nvestment result? The reason s that once you have a model for RAC, nevtable questons arse about the rsk/return contrbuton of the varous components of the model. Management wll qute rghtly want to know whch busness segments are contrbutng ther far share and whch are not. To push the RAC model of returns down to busness segment level requres a corporate-wde busness evaluaton system reasonably consstent wth the RAC methodology. It s easy to see that f such a busness evaluaton model s to properly evaluate underwrtng results, t must reflect some knd of nvestment ncome to be able to accurately evaluate the relatve values of the varous short and longer-tal busness segments. Many actuares consderng ths ssue have opted to reflect some knd of rsk-free nvestment ncome only. The thought s that addng n total nvestment ncome would unduly dstort the dstrbuton of underwrtng results because of the ncluson of too much nvestment rsk. It s thought that t s better to account for ths addtonal rsk elsewhere, n the nvestment model. But how should nvestment ncome be reflected n calendar year results? In order to have a good underwrtng evaluaton model, calendar year results must be modeled by frst modelng accdent year or polcy year results, so that premums, expenses and losses can be ted together for dfferent exposure perods. Rsk-free nvestment CARepaper\paperFN9.doc 17

18 returns are used to dscount all cash flows arsng from each accdent or polcy year to a sngle evaluaton date. These rsk-free returns may dffer for each year. For the calendar year result, the calendar year dscounted cash flows can reflect any changes n each accdent or polcy year s results durng the calendar year that were not expected at the begnnng of the calendar year. For the smplcty that arses from havng non-overlappng exposure perods, you mght choose to model accdent years nstead of polcy years, as we do. So now we are n the stuaton of modelng the dstrbuton of the dscounted change n result durng the next calendar year for each past accdent year, and also for the next accdent year. A stochastc model for premum and loss development s most helpful here. The premum model s smpler than the loss model except for the ncluson of retrospectve or other later premum adustments and payments. But snce the really sgnfcant premum changes rely upon changes n loss evaluaton, and can thus be modeled as a functon of the losses, let us concentrate upon the modelng of the losses. We want a stochastc model for accdent year losses and the calendar year ncurred loss development thereof. There are many such models n the actuaral lterature to choose from [ref ]. Whchever model s selected, we need a reasonably good parameterzaton n order to produce reasonably good answers remember the garbage-n-gospel-out syndrome? A reasonably good parameterzaton for the accdent year loss dstrbuton can be obtaned from hstorcal nformaton sutably adusted to future level by the followng steps. CARepaper\paperFN9.doc 18

19 (4.2) Loss dstrbuton modelng steps 1. Defne future potental maor catastrophes. 2. Flter the maor cats out of the hstorcal loss data. 3. Model the fltered, non-maor cat losses, adusted to future level. 4. Model the future maor cat scenaros. 5. Glue them back together. The maor cats are those large loss events whose presence or absence from the hstorcal loss data dstorts the estmaton of future loss occurrence or loss run-off potental. These maor cats can be ether natural or man-made catastrophes. Examples of realzed maor cats dstortng recent loss data are Hurrcane Andrew, the Kobe earthquake, asbestos and polluton clean-up. Dependng upon the company s nsurance portfolo, there may be others. Ths s an mportant pont: what consttutes a maor cat event for a partcular company depends upon the company s partcular nsurance portfolo. An example of a maor cat whose absence dstorts the hstorcal loss data, and thus the extrapolaton to the estmate of future loss potental, s the non-occurrence of an earthquake centered on the New Madrd fault. The future occurrence of ths event s beleved to be very possble and of very large loss potental, but yet there has not been an occurrence snce CARepaper\paperFN9.doc 19

20 Maor cats are very rare events that may be dffcult to dentfy and are certanly dffcult to quantfy. Yet these events are the key to understandng and measurng nsurance company underwrtng rsk. Let us separate the maor cat scenaro modelng nto two types: natural and manmade. Natural maor cats are easer to model than man-made. Much data exsts for smaller and medum-szed natural catastrophes, and data exsts for some larger events lke Hurrcane Andrew and the Kobe earthquake. Rensurers especally have devoted much tme and effort analyzng nsurance exposure to natural catastrophes, and there are some reasonably good, commercally avalable models to quantfy the exposure of any nsurance portfolo. The models seem to be dong a reasonably good ob estmatng loss severty. The man problem s the estmaton of loss frequency, the nverse of whch s sometmes referred to by underwrters as return perod. The more dffcult problem s the modelng of man-made maor cats. These are loss events arsng from the unknown rsks of technologcal, economc, legal and socal development. Ths development may nclude the expanson of nsurance coverage for what had been consdered to be busness rsks. If we take asbestos and polluton clean-up as canoncal examples, we can say that man-made maor cats have the followng characterstcs. CARepaper\paperFN9.doc 20

21 (4.3) Characterstcs of man-made maor cats Arsng from technologcal, economc, legal and socal development No sngle, well-defned event causng all the clams Long clams dscovery perods Unknown number of clamants No geographcal lmtaton Modelng man-made maor cats s very dffcult. There s great uncertanty regardng approprate model structures and realstc parameter values. Hstorcally, we have had so far only two maor events whch mght be consdered to be man-made maor cats: asbestos and polluton clean-up, and the ultmate nsurance losses from these two events are stll very much unknown [ref. 4.4]. But snce these man-made maor cats are so mportant to the evaluaton of future loss occurrences and loss run-off, and thus to the evaluaton of rsk adusted captal, we must do somethng. The modelng of these man-made maor cats s so complex and yet so mportant that t deserves a secton to tself. So we wll put off ths modelng dscusson untl the next secton. Let us temporarly assume that we have successfully modeled these man-made maor cats, and thus contnue the dscusson of the underwrtng model. The fltered loss dstrbutons may be modeled usng standard actuaral methods. We wll want to create the dstrbutons of the next calendar year ncurred losses arsng from each prevous or current accdent year and also the next accdent year. The standard methodologes tell us to segment the company s nsurance portfolo nto ts CARepaper\paperFN9.doc 21

22 maor components. Some of these components may be sngle contracts large enough and sgnfcant enough to be analyzed and modeled separately. In addton to lne of busness, a well-dversfed company may also need to consder geographc area. Sutable hstorcal exposure and loss data must be analyzed, model structures determned and parameters estmated. The models can then be extrapolated to the next calendar year. Note that ths extrapolaton tself ncreases uncertanty because of future nflaton and market rsks. Most actuares would say that ths ncreases parameter rsk. It s mportant to also model and estmate correlaton among the busness segments, both loss event correlaton and prcng or rate-level correlaton. The loss modelng obvously depends upon the exposure estmates by year. And let us not forget that the modelng of premums and expenses depends upon the ratng/underwrtng cycle and the degree of prcng or rate-level correlaton among the lnes. The model for the fltered losses s combned wth the models for natural and manmade maor cats to yeld the dstrbuton of gross ncurred loss for the next calendar year. Combne ths wth the model for premums and expenses and rsk-free nvestment ncome on the cash flow to obtan the dstrbuton of the underwrtng result for the next calendar year. Now let us return to the modelng of man-made maor cats. CARepaper\paperFN9.doc 22

23 5. Modelng Man-made Maor Catastrophes Before descrbng an approach to the modelng of man-made maor cats, we want to pont out the goal once more. Both the natural and man-made maor cats have a sgnfcant mpact on the company s result, affectng the tal of the loss dstrbuton, and thus the calculaton of RAC. But snce they occur wth such low frequency, they cannot be modeled approprately on the bass of hstorcal loss records only. Ths s especally true for the man-made cats. So nstead, we model them usng addtonal nformaton. Thnk of t ths way. Instead of usng only statstcal and actuaral methods to model the tal of the loss dstrbuton, we want to take nto account as much nformaton as possble. Ths nformaton comprses hard data such as past loss experence, portfolo nformaton and market nformaton, but also soft factors such as expert-knowledge and gut feelng. In addton, qute a bt of pure guesswork s necessary. Some mportant ssues related to the modelng of man-made maor cats are llustrated wth followng example. Suppose we ask ten experts to estmate the frequency of an ol tanker polluton event wth an mpact comparable or worse to the Exxon aldez accdent. We wll probably get more than ten answers! Ths s not only because of dfferent opnons among experts, but also because of a dfferent understandng of the queston: what do we mean by comparable mpact? Is t the mpact on the envronment, on the socety, on the economy or on the nsurance ndustry? Are we askng for the expected frequency n the next year, n the next 10 years or n the next 100 years? Are we askng for the frequency of ol tanker accdents only, or do we CARepaper\paperFN9.doc 23

24 want to nclude ol platforms as well? Are we askng for the frequency ust for Alaska or are we nterested n the worldwde number of smlar events? Because of ths ambguty, we must accept the fact that there cannot be a sngle correct answer. We wll always end up wth a set of possble solutons. Obvously, there s not only uncertanty related to the stochastc nature of the modeled events. The dstrbutons used to descrbe ths stochastcty are also uncertan. Ths s usually descrbed as model and/or parameter rsk. Whenever possble, ths uncertanty should also be ncluded n the overall RAC model. The tal of the loss dstrbuton to be modeled should represent all events whch mght have a maor fnancal mpact on the company. In order to derve a reasonably accurate dstrbuton, we want to quantfy the mpact (severty) and the occurrence probablty of all relevant events. Ths s mpossble. Therefore we must restrct the evaluaton to a lmted number of representatve scenaros. In dong so, we can exactly defne whch events we want to consder and whch assumptons we want to make. For each such specfc event, t s then possble to quantfy the fnancal mpact on the company (ths s a sort of stress testng). Much more dffcult s the estmaton of the occurrence frequency of the events represented by each representatve scenaro. We could of course also start wth the frequency of the events and then try to quantfy ther severty n a second step (what s the mpact of an event expected to occur wth a frequency of 0.01?). It should be clear that there s not a unque methodology for modelng man-made maor cats. The modelng method depends on the nature of the rsks, the lne of CARepaper\paperFN9.doc 24

25 busness, the avalable nformaton, etc. A model for man-made maor cats wll never be complete, but wll have to be adapted and modfed over tme n an ongong process. The modelng of future man-made maor cats nvolves underwrters, actuares, scentsts, engneers, clams people and fnancal analysts. The steps nvolved are as follows. (5.1) Modelng man-made maor cats Defne the characterstcs of a maor cat event Identfy potental maor cat exposures Estmate the frequency probablty of each maor cat event Estmate the severty dstrbuton of each maor cat event Translate all ths nto nsurance coverage We have already dscussed some defnng characterstcs of man-made maor cats. But let us restate them somewhat dfferently. (5.2) Characterstcs of man-made maor cats No sngle, trggerng event Unknown temporal duraton (development of loss exposure over many years) Unknown geographcal boundares (geographc development from local to global) CARepaper\paperFN9.doc 25

26 Lmted knowledge of event frequency and severty probabltes Lmted knowledge of nsured values because of unknown: - lnes of busness exposed - types of clams (BI, PD, fnancal) - number of clams, plantffs, nsureds - amount of compensaton per clam Lmted knowledge of the reacton of socety and nsurance markets (If losses are to be pad only n the future and f the ndustry can collect enough addtonal premums and f the losses need not be reserved too quckly, there may be less calendar year mpact.) The frst step n dentfyng the maor cat exposures s to hold branstormng sessons of underwrters, actuares, clams people, scentsts, engneers and fnancal analysts to lst scenaros whch mght generate maor cats. Let us call these scenaros threats. An example of such a lst mght look somethng lke ths. CARepaper\paperFN9.doc 26

27 (5.3) Potental threat scenaros (from branstormng sessons) Addtves (food and other) Anmal feed Asbestos Bo-gen technology Buldng materals Collapse of brdge or tunnel EMF Implants (medcal devces) Lead Polluton Surveyors Tobacco Alumnum Archtects/Engneers Banks BSE (mad-cow dsease) Chemcals Drectors and Offcers Food Lawyers/Accountants Pharmaceutcals RSI Terrorsm Transport of hazardous materal The next step s to elmnate threat scenaros whch are udged unlkely to fulfll the characterstcs of a maor cat event or whose lkelhood of occurrence s suffcently remote. The retaned dentfed threats are those whch have had some nsurance ndustry clams actvty. A pruned down lst mght look somethng lke ths. (5.4) Identfed threats Asbestos Buldng materals CARepaper\paperFN9.doc 27

28 EMF (electro-magnetc force) Implants Pharmaceutcal Polluton RSI (repettve stress nures) Y2K Our next step s to classfy these dentfed threats accordng to ndustry actvtes from whch they mght arse. (5.5) Threats and ndustry actvtes Threat Asbestos Buldng materals EMF Implants Pharmaceutcal Polluton RSI Y2K Industry Actvty Asbestos manufacturng, constructon, buldng Constructon, chemcals, buldng, owner/management Electroncs, power, ralroads, communcatons Medcal devces Pharmaceutcals All ndustres; especally chemcal, paper, machnery All ndustres; especally constructon, equpment Software and chp manufacturers, all ndustres D&O CARepaper\paperFN9.doc 28

29 The next step s to establsh a lst of trggerng events from a purely scentfc/techncal pont of vew, so that we would have some early warnng of possble occurrence. For each dentfed threat, we estmate ts degree of maturty. The next step s to determne whch nsurance lnes of busness the dentfed threats mght mpact. (5.6) Threats and nsurance lnes of busness Threat General Products Workers Lablty Lablty Comp. Asbestos yes yes yes Buldng materals yes yes yes EMF yes yes yes Implants no yes no Pharmaceutcals no yes no Polluton yes yes no RSI no yes yes Y2k yes yes no The next step s to estmate the frequency of occurrence for each threat. Snce none of the dentfed threats wll occur exactly as modeled, we must estmate the frequency for a larger set of smlar scenaros. As the probabltes mght change over tme, t mght also be necessary to specfy the consdered tme perod. As we have CARepaper\paperFN9.doc 29

30 seen and as we may guess, past experence s most lkely nsuffcent and s not necessarly representatve of future probabltes. Because of the long development perod nvolved, the loss probabltes are subect to technologcal, economc, fnancal, socal, legslatve and ursdctonal changes. The estmaton of nsurance loss severty s even more dffcult for all of the above reasons. In addton, we must analyze current and possble future nsurance coverage standard wordng regardng possble clam trggers, geographcal scope of coverages, clam seres clauses, cost n addton to lmts and aggregaton of lmts. The partcular nsurance portfolo, past and future, must be analyzed to dentfy exposed nsureds and coverages. We want to end up wth portfolo-wde loss frequency and severty probablty dstrbutons for each dentfed threat. For these events, we must remember that there s a strong correlaton among the losses over many accdent years. Ths s a bottom-up threat exposure analyss for a partcular nsurance portfolo. An alternate estmate for a partcular nsurance portfolo s a top-down market share assessment once a total or ndustry-wde estmate has been made of an dentfed threat s severty, For a very mmature threat, ths may be the best that can be done. Note that the estmaton procedures outlned above are the same as those used over the last 15 years to estmate asbestos and polluton losses. CARepaper\paperFN9.doc 30

31 6. How To Estmate RAC: Investment Model We now turn to the modelng of the dstrbuton of the company s calendar year nvestment result. We defne the nvestment result, CR I, as follows. (6.1) CR I = calendar year change n assets accounted for n RBC mnus tems already accounted for n CR U (such as premums, expenses, losses, rsk-free nvestment ncome calculated for underwrtng) mnus tems that wll be accounted for n credt rsk, CR C Snce we are modelng the change n a company s assets over a calendar year, the rsks that must be taken nto account are those arsng from the nature of the company s assets. (6.2) Asset Rsks Interest rate rsk Default rsk Stock market/equty rsk Real estate rsk Foregn exchange rsk Constrants upon nvestment polcy stemmng from nsurance regulaton n many ursdctons are a complcaton for nsurance companes. CARepaper\paperFN9.doc 31

32 The modelng of the dstrbuton of a company s calendar year nvestment result s a very complex and dffcult problem. We wll only be able to dscuss some concepts and some well-recognzed tools for attackng the problem. The nterested reader can fnd most of the detals n the references, but wll have to decde for hmself how exact the modelng should be. Ths modelng s closely related to the modelng of market rsks n the bankng world. However, there s one maor dfference: the tme horzon, as wll be dscussed below. Probably the best known tool for measurng market rsks s RskMetrcs TM by J.P. Morgan [ref. 6.1], also known as alue at Rsk (ar). There s a great amount of nformaton about ar on the Internet; a comprehensve lst of publshed and workng papers can be found n reference [6.2]. The descrpton of ar methodology n Appendx 13.1 s based on reference [6.3]. The general dea of ar s to combne a stochastc model for the basc rsk factors together wth a determnstc model that lnks the value of the fnancal nstruments n a portfolo to the random changes of the rsk factors. The dstrbuton of the change n value of the portfolo s obtaned by fttng a parametrc dstrbuton (very often a normal dstrbuton) to the frst two moments. ar s defned as a percentle (usually the frst percentle) of the result dstrbuton of the assets. However, we are nterested n the whole dstrbuton, not n sngle percentles. But snce the dstrbuton s defned by the set of all of ts percentles, t s possble to use ar-methodology to model the whole dstrbuton of nvestment rsk. However, there are several shortcomngs. CARepaper\paperFN9.doc 32

33 (6.3) Problems wth the Use of ar To Model the CR I The asset portfolo s actvely managed ar s a pont-n-tme concept ar assumes that market rsks can be modeled by means of a Markov process ar assumes complete lqudty of all nvestment nstruments ar s usually based on normal dstrbutons In contrast to the farly stable nsurance portfolo, an nvestment portfolo s more actvely managed over the one year tme perod we are consderng. Thus, management nteracton can nfluence the rsk. The second pont s that a fundamental assumpton of ar s that the tme horzon s short, one day f not nfntesmal, because some asset portfolos may not be constant n ther composton and may have tme dependent characterstcs. Snce we need the dstrbuton of CR I over a perod of one year, the queston s how to scale the dstrbuton to ths longer tme horzon. A problem s that the varous methods of scalng ar to dfferent tme horzons do not consder the nfluence of actve management. Thrdly, the assumpton that market rsks can be modeled by means of a Markov process, wheren the future probabltes depend only upon the current state, not upon prevous hstory, s somewhat questonable. Fourthly, t s clear that there s a great varaton n the lqudty of nvestment nstruments. Some are very lqud (e.g., stocks of bg publc companes or some standardzed dervatves), whle others take an CARepaper\paperFN9.doc 33

34 ntermedate poston (e.g., corporate bonds) and some are very llqud (e.g. mortgage backed securtes and especally real estate). Lqudty n general depends upon economc condtons. Ffthly, the assumpton of lnearty between changes of the rsk factors and changes of the values of the fnancal nstruments s an approxmaton only. And fnally, ar s usually based on normal dstrbutons. Snce we are nterested n the tal of the overall result dstrbuton, t mght be necessary to replace the normal dstrbutons by heaver-taled dstrbutons. To overcome these lmtatons, t s necessary to ncorporate the dynamcs of the portfolo management nto the model. We are not n a poston to offer a satsfactory soluton for ths problem. Instead, we wll only outlne the standard ar concept and dscuss a few approaches for extendng ths smple model. More detals on ar are n Appendx If a dffuson process wthout memory effects (.e. no seral correlaton) s used for modelng the changes of the rsk factors underlyng nvestment rsk, then all varances and covarances of the components of the nvestment portfolo wll ncrease lnearly wth tme. Because all varances and covarances are scaled wth tme, the portfolo varance s also scaled by the same factor. If the dstrbuton s also scaled wth the same factor, t follows mmedately that ar based on one day can be scaled to k tmes one day by multplyng t wth the square root of k. (6.4) ar for k days ar k days = ar 1 day k CARepaper\paperFN9.doc 34

35 The tme horzon to whch ar can be scaled should be related to the perod wthn whch no management acton s taken, or, n short, the holdng perod of the asset mx. However, our tme horzon s one year, thus contradctng the assumpton of a fxed portfolo. So, how can we estmate the portfolo component covarance matrx for a one year tme perod? One path s to construct ar from hstorcal tme seres. There are many dfferent ways of dong so. The sample mean of monthly data could be taken and be scaled to one year by takng k = 12. Usng daly data would result n k=250, ths beng the number of tradng days of a year (one could also take k=365 n argung that the rsk factors contnue to change over weekends, etc.). We could also estmate the covarance matrx by usng an exponental weghtng scheme. However, the resultng covarances would be much more responsve to the near past and would be more volatle over tme. Thus they mght not be a good bass for determnng the portfolo rsk over a one year tme horzon. Choosng a more sophstcated methodology for estmatng the behavor of rsk factors over one year does not relax the assumpton of havng no portfolo management acton takng place, but may yeld a better descrpton of the rsk factors. Another modelng and estmaton problem s lqudty. The vast unverse of securtes s very heterogeneous wth respect to lqudty. Lqudty may change due to ntroducton of new products supersedng others. Markets can get stcky because of general downturns and mtatve behavor. Lqudty measured as a dscount to ordnary prces can evaporate due to the sze of the partcular lot to be lqudated. CARepaper\paperFN9.doc 35

36 Bg lots often trgger wdenng spreads and thus ncur so-called market mpact costs. Because of ths, t would seem that the lqudaton perod s the maxmum tme perod that the ar model could be extended for a sngle nvestment nstrument. But, as dfferent nstruments have dfferent lqudaton perods, some means of fndng a proxy s needed. One obvous choce s a weghted average lqudaton perod. Another choce s to ncorporate the nstrument specfc lqudaton perod nto the exposure. Ths leads to some sort of exposure weghted tme horzon. Even f such an approach were adopted, gvng a better pcture of the today s loss potental, the problem of expandng the horzon to one year s stll not solved. The most dffcult modelng and estmaton problem s that of actve management. Over a one year tme perod, portfolo managers wll revse ther portfolo mx n order to realze opportuntes where they foresee them, condtonal on some restrctons, be they legal or other, and wth an eye on loss potental. Ther acton s amed at pushng the portfolo value up, or n extreme cases to stop further losses. A strategy s a set of rules, whch are appled when some nstance of the state varables materalze. We can thnk of ths as beng a mult-sequental two player game where a passve strategy of the envronment s confronted by the actve strategy of the portfolo manager. The envronment makes a move nto a new state of realty. The manager then makes a move by changng hs forecast and revsng hs portfolo. Dong nothng s also an acton. Wthn ths framework, the dstrbuton of the fnal value of the portfolo can only be assessed by smulaton. CARepaper\paperFN9.doc 36

37 The approach mplemented should be a relable model of realty, but also be pragmatc and ntellgble by a broad audence. The basc ar model s somewhat understood by many senor managers. Therefore, the ar model should serve as a startng pont. Ideally, the one-year extended ar model percentle would be a functon of the basc ar model percentle. The smplest way of dong so s a multplcaton rule of the followng form. (6.5) One-year extended ar model percentle ar 1year = g(θ) ar 1day 250 where g(.) s a functon of the strategy θ and the other consderatons dscussed above. Several ways of estmatng g(.) are pragmatcally possble. We could study hstory to fnd out what range g covered n the past. Another method would be to ntervew the portfolo managers regardng ther strategy, and then smulate the outcomes. Also, we could look at the gudelnes and predefned strateges n the case of an emergency. 7. How To Estmate RAC: Credt Model CARepaper\paperFN9.doc 37

38 We turn now to the modelng of the dstrbuton of the company s calendar year credt result. For convenence, we repeat and expand the lst of components of credt rsk noted n (3.1). (7.1) Credt rsk Rensurance ceded Accountng balances due Letters of credt Credt surety busness assumed Fnancal guarantee busness assumed Credt dervatves Etc. Note that we are ncludng here as credt rsk part of what may be thought of as part of the underwrtng rsk: credt surety and fnancal guarantee busness. The reason s that ths nsurance coverage s much more dependant upon general economc condtons and upon the fnancal condton of the nsured party than s the coverage for the more typcal underwrtng nsureds. Modelng the ceded rensurance component s ust lke modelng the underwrtng portfolo, except now we must account for the addtonal rsk of future non-collectblty CARepaper\paperFN9.doc 38

39 of clams payments by the rensurers. Ths ceded rensurance result s obvously hghly correlated wth the gross underwrtng result. In a very smple approach, the dstrbuton of the credt result could be modeled by combnng hstorcal results and threat scenaros n a manner smlar to the modelng of the underwrtng result as descrbed n sectons 4 and 5. However, a potental credt loss not only depends upon the credt exposure and the sze of some future random events, but also upon the fnancal strength of the counterpartes (debtors) nvolved,.e., upon ther default probabltes. Snce the default of several counterpartes can be trggered by a common event (e.g., a large earthquake, a recesson, etc.), there s potental strong correlaton wthn the credt portfolo and between the credt portfolo and the underwrtng and nvestment portfolos. Whenever possble, the credt model should combne all relevant rsk factors,.e., credt rsk should be modeled bottom up. Ths means that we should frst model the rsk of each component separately, and then, by modelng the correlaton of the components, we can evaluate the credt portfolo s overall rsk. In prncple we wll fnd that the deas drvng the modelng of the credt result are smlar to those dscussed for the underwrtng and nvestment models. Some credt models have so far been establshed n practce see references [7.1]- [7.6]. These all follow dfferent approaches and rely also upon dfferent assumptons. The selecton of an approprate model depends upon the purpose and the goals to be acheved. We cannot n ths paper lst all the advantages and dsadvantages of the CARepaper\paperFN9.doc 39

40 referenced models. Instead, we wll smply present some of ther common deas that drve the estmaton of credt rsk. Further detals can be found n the references. Snce we beleve that the credt rsk should be modeled bottom up, we wll dscuss the modelng of a sngle counterparty s credt rsk. We beleve that ths wll gve you some of the flavor of the modelng ssues. Later we wll very brefly dscuss the modelng of the whole credt rsk portfolo. But the full dscusson of ths very much more dffcult problem must be left to the above noted references. The credt rsk for a standalone transacton s prmarly dependant upon the counterpart credt qualty (ratng) and upon the contract type (ceded rensurance, fnancal guarantee, zero coupon bond, floatng rate, coupon bond, credt and surety, default swap, default dgtal, default opton, etc.). A smple model has three components. (7.2) Components of a smple model for a sngle transacton (contract) default frequency default exposure default loss severty Usually these components can be modeled ndependently and glued together at the end. Ths separaton s motvated by the fact that for smple contracts, these three components are almost ndependent. The default frequency usually depends ust upon the counterpart s credt qualty, the default exposure upon the type and lmt of the contract, and fnally, the default loss severty depends manly upon the senorty CARepaper\paperFN9.doc 40

41 of the contract. However, these three components are not necessarly ndependent for the case of ceded rensurance, snce the default probablty of an assumng rensurer and the default loss severty mght both depend upon the sze of a specfc catastrophc event (e.g., an earthquake n Calforna). And yet f we are clever enough n the constructon of our model, we may stll be able to use ths smple model for the default of a rensurance cesson. We wll now dscuss the modelng of these three components, wth specal attenton pad to the default probablty because t s the most senstve part of the ntegrated model. CARepaper\paperFN9.doc 41

42 (7.3) Two methods for estmatng the default frequency Ratng agency statstcs Black-Scholes-Merton model The default frequency corresponds to the probablty that the company wll default durng the specfed tme perod (n our case, one year). Ratng agences analyze the credt worthness of each company by lookng at ther cash flow, proftablty, fnancal flexblty, ndustry sector, market poston, compettors, management, controls, fnancal reportng and legal structure, etc.. Ths process s known ether as fundamental analyss or as the ratng process. Its outcome s a valuaton of the company s credt worthness. Both Moody s and S&P can estmate the default probablty of each ratng class over varous tme horzons based upon hstorcal data. The followng table shows default probablty percentages as a functon of ratng class at the begnnng of the tme perod and the length of the tme perod. CARepaper\paperFN9.doc 42

43 (7.4) Default frequency (probablty) by S&P ratng class [ref. 7.7] Term AAA AA A BBB BB B C Note that these default frequences are based upon hstorcal data. However, past default frequences may not accurately predct proect future frequences because of the presence of economc cycles. Thus our estmates of future default frequences must take nto account the current state of the economy. CARepaper\paperFN9.doc 43

44 (7.5) Black-Scholes-Merton Model Black-Scholes-Merton Model Assets Equty Debt Debt level Market asset value A A ' Dstrbuton of asset value at horzon Default Probablty Market alues Asset Market alues horzon (1 year) Current Market Debt alue Equty Market alue Current Equty Market alue Equty = Call opton on frm's assets wth strke prce the debt and maturty 1 year. Face Debt alue Current Market Asset alues Asset Market alue From the above pcture, we see that the equty market value of a company may be thought of as the prce of a call opton on the market value of ts assets, wth strke prce equal to ts debt (lablty) value. To evaluate the call opton prce, we can assume that the probablty dstrbuton of asset values at the end of the year satsfy certan Black-Scholes-Merton propertes, where the call opton prce can be calculated by the BSM-formula. (7.6) Black-Scholes-Merton formula Equty = BSM[underlyng spot prce (asset value); volatlty (asset volatlty); tme to maturty (1 year); rsk-free rate; strke prce (debt)] CARepaper\paperFN9.doc 44

45 In case we do not know the volatlty, we look for the market prce of a call opton and then nvert the BSM-formula. Smlarly, snce we may not be able to drectly observe nether asset volatlty nor asset prces, we can calculate them from market equty prces, nvertng the BSM-formula and usng the delta hedgng formula. If we can assume that asset returns are normally dstrbuted (mean = actual asset value, volatlty = BSM mplct volatlty), then we can derve the probablty that the asset value falls below the threshold of debt-lablty value. Ths probablty s defned to be the expected default frequency (EDF). Note that n ths model, EDF s derved from the equty market value, whch s based on nvestors future expectatons about the company. The second component of the sngle transacton model s the default exposure. The default exposure depends upon the partcular contract and not much upon a company s credt qualty. For the case of ceded rensurance, the credt rsk default exposure s gven by the amount of loss ceded to the rensurance contract and the shares underwrtten by the varous rensurers. Default exposure patterns are usually assumed to be flat (= average default exposure) over the tme perod, except for contracts that have a long duraton. We must remark here that ths strong assumpton does not hold for dervatve products. Ther default exposure can ncrease or decrease dependent upon market movements. For these dynamc default exposures, we can calculate a default exposure envelope. Ths s an estmate of the default exposure probablty dstrbuton durng the tme perod. We must also remark that collateral or nettng CARepaper\paperFN9.doc 45

46 agreements can substantally reduce default exposure. The modelng of these s straghtforward, but adds consderable complcaton. The thrd component of the sngle transacton model s the default loss severty. The default loss severty s the percent of the total owed whch s not pad followng a default. Often t s more convenent to look at the complement, 1 {default loss severty}, whch s the recovery percentage. Default loss severty manly depends upon the contract structure (senorty), and usually less upon the ratng. A standard model calculates the probablty dstrbuton of the recovery va a beta dstrbuton ft to hstorcal data wth the key nput varables beng the contract senorty and ratng. We fnally observe that the default loss severty usually does not depend on the duraton (tme perod). Let us assume that we have estmated the three model components for a sngle transacton. We can now put them together. Snce these three components are all ndependent, we smply convolute ther dstrbutons. The standalone captal needed for a sngle transacton then corresponds to a percentle (for example, 99%) mnus the expected default loss, whch s calculated by multplyng the expected values of the components. (7.7) Expected default loss Expected default loss = EDF E[default loss severty] E[default exposure] CARepaper\paperFN9.doc 46

47 The more dffcult problem s to step up from the modelng of sngle transactons to the modelng of a whole portfolo of transactons. To do so, we must estmate the correlaton between sngle transactons. For the case of ceded rensurance, some correlaton between varous ceded rensurance contracts can be accounted for va threat scenaro modelng as n secton 5. For each scenaro, we can estmate the percentage of loss whch wll not be recovered (the default loss severty) from each rensurer. Even though the modelng of the rsk of the whole credt portfolo s the really dffcult and more nterestng modelng problem, addressng t would requre a substantally longer paper. Thus we must leave ths to the nterested reader, who can fnd many good deas n the references by Tom Wlson, [ref ]. An analytcal procedure for calculatng the aggregate loss dstrbuton s too complcated (especally for a portfolo contanng hundreds of contracts). The only possblty for estmatng the fnal loss dstrbuton s the use of Monte-Carlo smulaton. 8. How To Estmate RAC: Puttng It Together The last step n the estmaton of RAC s the constructon of the dstrbuton of CR, the company s calender year result. Once we have an estmate for the dstrbuton of CR, the management rsk tolerance rule defnng RTL wll produce a value for RAC va (3.9). CARepaper\paperFN9.doc 47

48 Remember that CR s smply the sum of the underwrtng, nvestment and credt results (3.6). After dong the hard work of constructng the underwrtng, nvestment and credt result models n sectons 4-7, puttng them together s almost antclmactc. The problem here s to account for correlaton and other hgher moment dependences among the components. As already dscussed, the ceded rensurance part of the credt result s hghly correlated wth the underwrtng result. We have no magc solutons to offer to ths hghly complex problem. Agan we wll suggest some smple methods. The problem s how to construct the dstrbuton functon for CR from the models for the underwrtng, nvestment and credt results together wth nformaton about ther nterdependences. We assume that our modelng of the underwrtng, nvestment and credt results also produces nformaton about the correlaton matrx for these results, but that we have no nformaton about hgher moment dependences t s dffcult enough to estmate correlaton, much less hgher moments. So our nformaton s not suffcent to determne the common overall result dstrbuton, snce the nformaton regardng hgher moments s ncomplete. In prncple, wth reasonable assumptons, the dstrbuton of CR can be obtaned analytcally. In practce however, ths s often too complcated, so smulaton s used to generate the dstrbuton of CR from the component dstrbutons together wth ther correlaton. Thus we have modelng alternatve 1. (8.1) CR Modelng Alternatve 1: Smulaton usng the component dstrbutons and ther correlaton CARepaper\paperFN9.doc 48

49 In the followng, we wll present some alternatve approaches to the constructon of the dstrbuton functon for CR. These methodologes permt analytcal constructons. The frst s an old actuaral standby. (8.2) CR Modelng Alternatve 2: Fttng a selected parametrc dstrbuton va the Method of Moments The smplest modelng approach s to determne the frst two moments of CR from the moments and correlatons of the components, and then ft an approprate parametrc dstrbuton va the Method of Moments. Ths s the methodology used for modelng asset rsks va the ar methodology as descrbed n secton 6 and Appendx The mssng nformaton wth respect to hgher moments s flled n va the selecton of the dstrbuton type for CR. Snce the ratos (percentle mnus expectaton)/standard devaton strongly depend on the shape of the dstrbuton, the selecton of an approprate type of dstrbuton s most mportant. The decson of whch dstrbuton type to use must be based on the shape and relatve mportance of the dstrbutons of the components. For the case of a rensurance company, the underwrtng result dstrbuton s almost always heavly skewed. Ths wll most lkely cause the dstrbuton of CR to be also skewed as we pctured n (3.7). (8.3) CR Modelng Alternatve 3: Buldng a herarchcal stochastc compound model based upon common underlyng rsk factors A slghtly more complcated modelng approach s to buld a model based upon common underlyng rsk factors. Ths knd of herarchcal stochastc model s called a CARepaper\paperFN9.doc 49

50 compound model. In a compound model, determnstc event frequences are replaced by a stochastc varable. For example, suppose n the underwrtng model that the dstrbuton of the number of losses (or events) N wthn each model component depends upon a loss frequency parameter λ, and the dstrbuton of the each parameter λ depends upon a common random varable χ n the followng way. (8.4) λ = λ 0, χ wth E[λ ] = λ 0, where χ s a random varable wth E[χ] = 1 Then all the component models wll be correlated va χ. Ths approach can of course be extended by ntroducng a set of common rsk factors as outlned n sectons 6 and 7 for the case of the nvestment rsk model and the credt rsk model, respectvely (see also Appendx 13.1). The loss frequency of each component model s then gven by a lnear combnaton of the fundamental rsk factors. Clearly, ths type of modelng can be extended to all three components of CR smultaneously n order to buld a compound model for the dstrbuton of CR. (8.6) CR Modelng Alternatve 4: Constructng Loss Frequency Curves Our last approach s especally useful for modelng correlatons for maor cat events, whether they cause underwrtng, nvestment or credt losses or any combnaton of these types of loss. Snce RAC s determned by maor cat loss events of whatever type, we can use ths modelng approach to determne the left hand tals of the results CARepaper\paperFN9.doc 50

51 dstrbutons va the rght hand tals of the potental loss event dstrbutons. In ths modelng approach, the dfferent model components can be nterpreted and related to observable quanttes. However, ths model s not restrcted to catastrophe-lke processes; t can be appled n many cases where the stochastc process s descrbed by a compound model [ref. 8.1]. The problem wth the use of ths model wll be the dffculty of transformng the annual aggregate models for nvestment and credt rsk nto the framework of compound models wth event frequences and severtes. If ths s possble, then ths model can be used to construct the dstrbuton of CR. In a compound model, the number of loss events and the ndvdual event loss severty are modeled separately. For the loss frequency curve model, nstead of representng the loss frequency λ 0 and the loss severty F(x) ndependently, we can use the loss frequency curve (LFC) for a ont representaton. CARepaper\paperFN9.doc 51

52 (8.7) Loss frequency curve (LFC) Frequency: E[N] = λ 0 and ar[n] = Q λ 0 Severty: LFC: cdf F(X) λ(x) = λ 0 (1 F(x)) Inverse LFC: λ -1 (λ 0 y) = F -1 (1 y) for 0 < y 1 Snce F(x) s the probablty that the severty of an loss X s less than or equal to x, then λ(x) s the expected number of losses above the threshold x. Conversely, gven an LFC λ(x), then the correspondng loss frequency λ 0 = λ(0) and the loss severty dstrbuton F(x) = 1 - λ(x)/λ(0) are easly constructed. We ntroduce three fundamental operatons on LFC s. (8.8) Operatons on LFC s: 1) Aggregaton n frequency drecton The ont LFC of two ndependent subportfolos s gven by the sum of the LFC s. In ths case, the expected number of losses above a certan threshold x s smply the sum of the frequences: λ(x) = λ 1 (x) + λ 2 (x) In the case of Posson loss frequency, the aggregaton n the frequency drecton s equvalent to convoluton. Instead of usng usng Paner recurson to calculate the dstrbutons of the component dstrbutons, we can smply add the LFC s and Panerze the aggregated LFC. In the non-posson case, the stuaton s more complcated. If the two LFC s may be calculated va Paner recurson, but Q 1, then CARepaper\paperFN9.doc 52

53 the ont dstrbuton wll generally not belong to the Paner class. However, t s stll possble to fnd a representatve from the Paner class fttng the frst two moments. (8.9) Operatons on LFC s: 2) Aggregaton n loss drecton The strongest correlaton between two subportfolos s obtaned n the case of comonotoncty. Then, the losses are determnstc functons (gven by the nverse LFC s) of a sngle random varable. In ths case, the LFC s are aggregated n loss drecton: λ -1 (y) = λ 1-1 (y) + λ 2-1 (y) for 0 < y λ 1,0 = λ 2,0 Comonotoncty means that two portfolos are affected by the same events,.e., Q 1 = Q 2 by defnton. If the expected numbers of losses are dfferent n the two subportfolos, we can always add an approprate number of zero -losses to the portfolo wth the smaller frequency n order to acheve λ 1,0 = λ 2,0. (8.10) Operatons on LFC s: 3) Frequency splt Ths s the opposte operaton of aggregaton n frequency drecton. If the losses observed n a subportfolo can be separated nto two ndependent classes, then we can construct LFC s for each class n such a way that the sum of these LFC s equals the total LFC. We have now seen how to aggregate ndependent and fully dependent (comonotonc) dstrbutons. Wth the help of the three fundamental operatons, we can construct any correlaton between two subportfolos as follows. CARepaper\paperFN9.doc 53

54 (8.11) Combnng two correlated subportfolos Splt the LFC of each sub-portfolo accordng to (8.10) nto a local noncorrelated component and a global correlated component. Aggregate the two subportfolo global components n loss drecton accordng to (8.9) to obtan an overall global component. Aggregate the overall global component and the two local components n frequency drecton accordng to (8.8) to obtan the combned LFC. Calculate the overall loss frequency and loss severty from the overall LFC, and calculate the overall loss frequency varance. The overall aggregate loss dstrbuton s then obtaned va Paner recurson. The model thus dstngushes between events affectng one of the subportfolos only ( local events) and events affectng both subportfolos at the same tme ( global events). For the case of two subportfolos, t s suffcent to ntroduce one group of global events. For modelng the correlaton structure between more than two subportfolos, ths smple procedure can be generalzed by the ntroducton of several global event groups. (8.12) Example 1: Excess layers The concept of comonotoncty s best known to rensurers wrtng shares of several layers of a nonproportonal rensurance program. The loss amounts CARepaper\paperFN9.doc 54

55 ceded to dfferent layers are determnstc functons of each ground up loss. The overall loss to be pad by the rensurer s the sum of all these ceded layer losses. The rensurer s LFC s obtaned by aggregaton n loss drecton. Losses affectng only the frst layer are zero losses for the second layer, whle losses affectng the second layer are total losses for the frst layer. (8.13) Example 2: Natural catastrophes In the case of a natural catastrophe, each nsurance company operatng n the affected area wll suffer a loss whch s very nearly determned by ts market share tmes the total loss. In ths case, a good approxmaton for the market LFC s obtaned by the aggregaton of the company specfc LFC s n loss drecton (comonotoncty). (8.14) Example 3 Wndstorms n Europe usually affect not ust a sngle country, but several countres at the same tme. The LFC for a rensurer s contnent-wde wndstorm exposure thus strongly depends on ts exposure n each country (or regon). The LFC s per country (regon) can be obtaned by scalng the correspondng normalzed market LFC wth the rensurer s exposure n the country. The overall LFC can then be constructed as follows: Determne the specfc LFC s for all relevant countres: e.g., for France, the Unted Kngdom, Benelux, Germany, Denmark etc. CARepaper\paperFN9.doc 55

56 Defne a set of event-groups: e.g., Europe (all countres); North (countres n northern Europe); South (countres n southern Europe); Contnent (Europe wthout Unted Kngdom). The LFC s per country are splt nto LFC s for local events and LFC s for each event group. The number of events per group can be derved from the analyses of the wnd felds of hstorcal storms. All LFC s belongng to the same event group are aggregated n loss drecton (comonotoncty) n order to obtan the overall LFC per event group. The overall LFC s per event group and all local LFC s are aggregated n frequency drecton to obtan the overall LFC. The model ntroduced above s best suted for modelng natural catastrophes where the assumpton of comonotoncty s best fulflled. However, clearly t can also be appled to the modelng of man-made threats. However, as we dscussed above, n order to use ths model for calculatng the dstrbuton of CR, the problem wll be the modelng of the nvestment and credt results va compound models. 9. How To Allocate RAC To Lne, Product, etc. CARepaper\paperFN9.doc 56

57 Once a company has estmated ts overall RAC, t s useful to allocate t down to varously defned rsk subportfolos (lne, product, proft center, etc.) of the overall rsk portfolo for the followng reasons. (9.1) Reasons to allocate RAC to rsk subportfolo Measure performance on a rsk-adusted and consstent bass Determne rsk-adusted proft margns for nsurance product prcng Evaluate alternate busness strateges on a rsk-adusted bass These three tems are not ndependent of each other. To encourage consstent decson-makng, t s desrable to have a consstent busness measurement and evaluaton structure. One way of accomplshng ths s to allocate RAC (or a smlar rsk measure) down to ndvdual busness unts: proft centers, lnes of busness, products and even ndvdual contracts. In ths secton, we dscuss methods for dong so. There s no general answer to the queston of how RAC should be allocated. We wll dscuss varous crtera whch mght be desrable for an allocaton method, and then evaluate varous suggested allocaton methods accordng to these crtera. In fact, allocaton of RAC s not necessary to accomplsh (9.1). But we wll see that a RAC allocaton defnes a coherent structure whch can make t easer to see what s gong on and encourage consstent decson-makng. The RAC allocaton s a vrtual allocaton, not a physcal or legal allocaton, ust as RAC tself s a vrtual nternal requred captal to support rsk. But snce RAC s CARepaper\paperFN9.doc 57

58 defned n relaton to the determnaton of avalable rsk bearng captal, RBC, dfferent corporate structures may gve rse to dfferent overall RAC, and may lead to dfferent allocatons. One beneft of a RAC allocaton s that t puts us n a poston to allocate an overall corporate return on equty (RoE) target to rsk subportfolos n a rsk-adusted and consstent manner. However, snce the overall RAC may be very dfferent from the equty used n the RoE denomnator (usually one of the publcly-perceved captals n (1.1)), the RoE target must be transformed nto a RoRAC (Return on RAC) target. If we can properly allocate down the RAC and RoRAC target to rsk subportfolos, t s clear we have the frst part of (9.1); we can measure subportfolo performance on a rsk-adusted and consstent bass. A tool that compares actual results wth target results s a powerful nstrument n the hands of controllers and management. But management acton s then requred to affect the future busness mx and future results of the company. Therefore, management must decde how to steer the company wth the help of ths tool, as we shall see n secton 10. Only after the overall busness goals (what should be optmzed) and the steerng process are clearly defned, can we decde how to allocate the overall targets to rsk subportfolos n order to best support the steerng process. Management must also decde how to translate the allocaton of RoRAC nto rsk-adusted proft margns for nsurance product prcng n order to meet corporate obectves. You mght ask: Why not accomplsh the RAC allocaton by calculatng the standalone RAC for each subportfolo? The problem wth ths method s that t gnores any CARepaper\paperFN9.doc 58

59 dversfcaton benefts created by combnng the varous rsk elements. Dversfcaton here means that the RAC for the overall rsk portfolo s less than the sum of the RACs of the ndvdual subportfolos. Ths s true for our smple RAC n secton 3 based upon a percentle (unless the subportfolos are completely stochastcally dependent) because the results dstrbuton of the portfolo s more compressed n a relatve sense than are the results dstrbutons of the varous subportfolos. Theoretcally, the percentles of the subportfolos are not necessarly sub-addtve, but n practce they usually are. The dversfcaton effect s greater the more the subportfolos are stochastcally ndependent. The varous subportfolos dffer n the degree to whch they ncrease the dversfcaton of the overall portfolo. Thus t s mportant to see how each subportfolo fts nto the total; ts contrbuton to the overall RAC may be less (or more) than s ndcated by ts standalone RAC. Another queston arses: If we have an overall RAC model, can we use a dfferent model for the RAC allocaton? Our smple RAC calculaton n secton 3 was based upon a percentle of the company s calendar year result. If we use a dfferent method for allocatng RAC, we must check whether or not the overall goals of the company can stll be met. We turn now from talkng about the partcular, smple RAC calculaton method descrbed n secton 3 to the general problem of allocatng any overall RAC down to subportfolo. Before decdng whch specfc method should be used for RAC allocaton, we should want to defne varous propertes that mght be fulflled by any allocaton method. CARepaper\paperFN9.doc 59

60 Some desred propertes can be derved from mathematcal prncples; others depend on economc, organzatonal or practcal consderatons. The fnal selecton of propertes that are most mportant s possble only n the context of management s overall goals and ther steerng process. The followng lst of propertes of allocaton methods does not represent a set of consstent axoms. Instead, the lst descrbes varous crtera whch we mght want the RAC allocaton to satsfy. The lst s not complete or mutually exclusve: some propertes exclude each other and some are already contaned n others. CARepaper\paperFN9.doc 60

61 (9.2) Propertes of allocaton methods Rsk adusted: The RAC allocated to a subportfolo accounts for the rskness of the subportfolo, as seen from the company s overall perspectve. Partalty: If two subportfolos share exactly the same rsk elements (contracts, equtes, etc.), X% for the frst, Y% for the second, then the RAC allocated to the frst and second s exactly n the proporton X to Y. Lnearty: The RAC allocaton s addtve; ths also means that the RAC allocated to a partcular subportfolo doesn t depend upon whch larger subportfolo t s contaned n. If ths crteron were not fulflled, there would be arbtrage opportuntes wthn the company. Account for dversfcaton and dependency: The RAC allocaton accounts for dependences among rsk subportfolos as well as ndependence. Organsatonal ndependence: The RAC allocated to a partcular subportfolo does not depend upon ts partcular organzatonal structure nor upon the organsatonal structure of the rest of the company (ths s closely related to lnearty). Measure dependent: The RAC allocaton s the nverse of the aggregaton process used to determne the overall RAC. Ths crteron s fulflled f the RAC allocated to each subportfolo s based upon the same underlyng rsk CARepaper\paperFN9.doc 61

62 model as s used for the overall portfolo, and f t also takes nto account dependences and dversfcaton n an approprate way. Based upon relable nformaton: The RAC allocaton s based on avalable and relable nformaton as far as possble. Practcalty: It s possble to mplement numercally n such a way that the RAC allocaton s stable and relable, and can be obtaned wthn a reasonable tme and wth ustfable effort. Reflect nternal rsk percepton: The RAC allocaton s based on nternal rsk percepton only. Ths may be n contrast to the overall RAC, whch mght be composed of nternal (percentle) and external (RTL) rsk percepton, as n our smple case. Ex post addtvty: The sum of the allocated subportfolo RAC (on the bass of ex ante nformaton) s equal to the overall ex post overall RAC for the entre portfolo. Consstency wth statutory requrements: The RAC allocaton takes statutory requrements nto account. Ths s qute a lst of possble propertes. No sngle RAC allocaton method can satsfy all of them. When dscussng allocaton methods, we would frst lke to dstngush between three dfferent classes. The frst class conssts of those allocaton methods that are based on local rsk measures. The overall RAC s allocated n proporton to the ndvdual CARepaper\paperFN9.doc 62

63 rskness of each rsk subportfolo as measured by the local rsk measure wthout takng nto account dependences among the subportfolos and dversfcaton. (9.3) Examples of Local rsk measures Standard devaton arance Percentles Shortfall rsk (condtonal expected excess loss) Usng local rsk measures wthout consderng the actual dependences between subportfolos does not mean that such dependences are rrelevant for allocaton. For example, the standard devaton prncple assumes full correlaton between all subportfolos; the varance prncple assumes ndependence; the percentle and the shortfall prncples assume comonotoncty. The second class conssts of those allocaton methods that are based on some knd of volume measure, such as solvency ratos. These methods play a specal role snce the local volume measure (as used by the frst class of measures) equals the contrbuton to the overall volume (property of the methods belongng to the thrd class). Ths s due to the fact that expected values are addtve and cannot be dversfed. The thrd class conssts of those allocaton methods whch are based upon the contrbuton of each subportfolo rsk to the overall global rsk. The overall RAC s CARepaper\paperFN9.doc 63

64 allocated n proporton to the contrbuton of each subportfolo to the overall rsk. Thus, these methods take nto account dependences as well as dversfcaton (as far as modeled). (9.4) Examples of Global rsk measures Margnal prncple (varaton, or wth and wthout as defned n Appendx 13.2) Euler -prncple (specal case of margnal prncple to be descrbed below and n Appendx 13.2) Covarance ( Euler for standard devaton) Hgher co-moments ( Euler for hgher moments) Lnear combnatons of above A RAC allocaton based upon the margnal prncple (as defned n Appendx 13.2 and not to be confused wth the smlar margnal change allocaton as defned below) or on the Euler prncple depends on the defnton of an overall rsk measure, snce these methods quantfy the contrbuton of each subportfolo to the overall rsk. So, let us defne rsk measure. (9.5) Rsk Measure: A rsk measure ρ s a mappng from the space of volume measures of subportfolos of a rsk portfolo nto the postve real numbers. We assume that there exsts a volume measure assocated wth each subportfolo R. For any subportfolo R, ρ( ) s the RAC allocated to R. CARepaper\paperFN9.doc 64

65 The Euler prncple allocates RAC by reversng the aggregaton process. Ths s acheved by answerng the followng queston: What s the contrbuton of each subportfolo to the overall RAC? The answer s of course trval for those components n the RAC formula whch are strctly addtve - as s the RTL n secton 3, snce t s based on solvency ratos. But what s the contrbuton of those RAC components whch depend on dversfcaton, such as percentles? Fortunately, there exsts a very general and conceptually easy soluton to ths problem. Snce the basc mathematcal prncple behnd t was found by the famous Leonhard Euler more than two centures ago, we call t the Euler prncple. (9.6) Euler Prncple: ρ( ) = {( / )ρ()} where ρ s a homogeneous rsk measure dfferentable on the space of volume measures of subportfolos R of the portfolo R, {R 1, R 2,... R n } s any partton of R, and s the volume measure of R. There s a more thorough development of ths concept n Appendx The margnal prncple s an approxmaton for Euler. The covarance prncple s a specal case of the Euler prncple (obtaned for the case where the overall rsk s defned by the standard devaton, as shown n Appendx 13.2). The followng specal captal allocaton method, also belongs to class 3, and s smlar but not dentcal to the margnal prncple wth and wthout. CARepaper\paperFN9.doc 65

66 (9.7) Margnal Change Allocaton : The RAC allocated to a subportfolo s gven by the margnal change of the overall RAC as a consequence of addng the subportfolo to the pre-exstng portfolo excludng ths subportfolo. Ths allocaton method has the feature that the overall RAC always equals the sum of the RAC s allocated to the subportfolos n the portfolo (ex post addtvty). However, t has a very serous shortcomng: the allocated RAC depends upon the orderng of the rsks; the dversfcaton between a new subportfolo and the exstng portfolo s fully attrbuted to the new subportfolo. Thus there s no clear rule on how to treat exstng rsks n a mature portfolo. Also, the mplementaton of ths method requres a very powerful and fully ntegrated onlne rsk modelng tool. In the followng table, we comment upon the propertes that one mght want fulflled by an allocaton method. We omt the property consstency wth statutory requrements, snce ths must be determned locally. CARepaper\paperFN9.doc 66

67 (9.8) Propertes of allocaton methods Rsk adusted Consder contrbuton to overall rsk Partalty Allocate the same RAC for the same rsk Lnearty RAC allocated s the sum of the ndvdual RAC s Dependency Dependences between rsk are consdered Organsatonal ndependence Allocated RAC does not depend on the structure of the company Local rsk measures (class 1) olume measures (class 2) Global rsk measures (class 3) not fulflled not fulflled fulflled Consders only full olume alone Subportfolo rsk s correlaton between does not contan measured as part of subportfolos or no nformaton about the overall rsk correlaton at all rskness partly fulflled fulflled fulflled OK for most In some cases (e.g., methods, but, e.g., margnal prncple) not not for varance! exactly not fulflled fulflled fulflled In some cases (e.g., margnal prncple) not exactly not fulflled fulflled fulflled olume cannot be dversfed not fulflled fulflled fulflled In some cases (e.g., margnal prncple) not exactly CARepaper\paperFN9.doc 67

68 Local rsk measures (class 1) olume measures (class 2) Global rsk measures (class 3) Measure not fulflled fulflled fulflled dependent Reflects the overall rsk measure If overall rsk s defned by volume Informaton Based on all relable nformaton partly fulflled Informaton regardng dependences s not consdered fulflled If overall rsk s defned by volume partly fulflled Hgher co-moments (for whch only lttle nformaton s avalable) may be mportant Practcalty fulflled fulflled partly fulflled OK for covarance, dffcult for others Reflect nternal rsk percepton Be derved from overall dstrbuton and not from volume fulflled not fulflled partly fulflled OK for hgher comoments, but not necessarly for Euler and margnal Ex post addtvty The sum of the RAC allocated to busness unts always equals the overall RAC not fulflled fulflled The expected value cannot be dversfed partly fulflled Exactly fulflled by one specal method, almost fulflled by other methods 10. Managng RAC To Optmze Rsk and Return CARepaper\paperFN9.doc 68

69 As you may recall from secton 3, we defned RAC to meet certan crtera, whch we repeat here for convenence. (10.1) Crtera for RAC It must meet specfed management rsk and survval crtera. It should quantfy the rsk/return trade-off for all rsk exposures. It must be useful for makng approprate rsk-based busness decsons. The frst crteron s part of the defnton of RAC n secton 3. In sectons 4-9, we dscussed the quantfcaton of the rsk/return trade-off va the estmaton of RAC and RoRAC for each segment of an nsurance company s rsk portfolo and the calculaton of the company s underwrtng, nvestment and credt results. In ths secton, we wll deal wth the use of RAC and RoRAC for makng busness decsons about the company s underwrtng, nvestment and credt portfolos. Among the many goals one mght have for the use of RAC and RoRAC to optmze rsk and return are the followng. (10.2) Some goals Acheve captal effcency Maxmze shareholder value Acheve a target RoE CARepaper\paperFN9.doc 69

70 Two partcular strategc actons we wll explore are the followng. (10.3) Two possble strategc actons Move RAC or RBC toward equalty Maxmze RoE wth respect to management s rsk tolerance rule Let us consder the frst strategc acton. (10.4) If RAC > RBC, then move toward RAC = RBC by one or more of the followng tactcal actons: Decrease underwrtng rsk Cede more rsk to (secure) rensurers Decrease nvestment rsk Decrease credt rsk - buy more secure rensurance (buy dversfcaton) Dvest non-core and RAC-ntensve operatons Rase new captal Ths s the case where the avalable captal, RBC, s not enough to properly support the company s rsk portfolo and also meet management s rsk tolerance rule. Thus the company s captal s exposed to more rsk than desred. The frst fve tactcal actons decrease RAC; the last ncreases RBC. CARepaper\paperFN9.doc 70

71 Note that buyng rensurance decreases underwrtng and also nvestment rsk (f t s not placed on a funds-wthheld bass). As dscussed earler, rensurance transforms an underwrtng rsk, and usually also an nvestment rsk, nto a credt rsk. The credt rsk s the possble non-payment or decreased payment of ceded clams caused by a fnancal default by the rensurer. The cost of ths credt rsk must be thought of as part of the cost of rensurance. If the rensurance s secure, the company s overall rsk level decreases. The sum of the company s and rensurer s RAC may be less after the placement of rensurance. Ths can occur f the rensurer s better dversfed than the cedng company, or f the assumed rensurance s less correlated wth the remander of the rensurer s portfolo than t s wth the cedng company s. In these cases, the rsk transfer causes the cedng company s RAC to decrease more than the rensurer s RAC ncreases, f ther management rsk tolerance rules are the same. An nternatonal rensurer may be better dversfed than a cedng company n the followng ways. (10.5) Well-dversfed rensurers typcally have better dversfcaton by Number of nsureds (smaller shares of many) Types of rsks (a broader underwrtng portfolo) Concentraton of rsk, e.g., geographcally Let us return to the frst strategc acton of movng RAC or RBC toward equalty. The alternate case s as follows. CARepaper\paperFN9.doc 71

72 (10.6) If RAC < RBC, then move toward RAC = RBC by one or more of the followng tactcal actons: Increase underwrtng rsk expand nto new busness wrte hgher rsk busness develop new products cede less rensurance Increase nvestment rsk Increase credt rsk Buy another company Return captal to shareholders Ths s the case where the avalable captal, RBC, s more than necessary to support the company s rsk portfolo and also meet management s rsk tolerance rule. Thus the company s captal s not beng fully utlzed to maxmze the RoE wth respect to rsk level. The frst four tactcal actons ncrease RAC; the last two decrease RBC. Now let us explore the second strategc acton of maxmzng RoE wth respect to management s rsk tolerance rule. The condtons to do so are clear. CARepaper\paperFN9.doc 72

73 (10.7) Maxmze RoE wth respect to management s rsk tolerance rule Smultaneously acheve the followng: Acheve RAC = RBC Maxmze RoRAC RoRAC, as we have seen, s smply the company s result dvded by the company s RAC. (10.8) RoRAC = CR / RAC If RAC = RBC, and RoRAC s maxmzed, then t s clear that the RoE s maxmzed n a manner satsfyng management s rsk tolerance rule. There may be, and probably s, more than one way to acheve (10.7). What we wll descrbe here s an teratve process. Our startng base case s the assumpton that the company s current underwrtng, nvestment and credt rsk portfolos wll contnue through next year. (10.9) Base Case: Current rsk portfolo Rsk portfolo components ndexed va = 1, 2,..., n. CR 0 [] = rv for next year s result for component CR 0 = Σ CR 0 [] = overall result for base portfolo RBC 0 = RBC for base portfolo calculated va secton 2 RTL 0 = rsk tolerance level for base portfolo CARepaper\paperFN9.doc 73

74 RAC 0 = RAC for base portfolo calculated va (3.9) E[RoRAC 0 ] = E[CR 0 ] / RAC 0 va (10.8) RAC 0 [] = standalone RAC for component ARAC 0 [] = overall RAC 0 allocated to component va secton 9 We need the dstncton between standalone RAC and allocated RAC. Now compare the expected return E[CR 0 []]/ ARAC 0 [] for each component to E[RoRAC 0 ]. Let us frst consder the case where overall RAC 0 < RBC 0. Let us assume that management has no wsh to decrease RBC, but nstead wshes to ncrease RAC. The frst possblty for component s that ts expected return dvded by allocated RAC exceeds the overall expected RoRAC. Component s thus the knd of assumed rsk we would lke to have more of. (10.10) Case 1: RAC 0 < RBC 0 and E[CR 0 []] / ARAC 0 [] > E[RoRAC 0 ] Then ncrease RAC[] by ncreasng the rsk level of the th component. The rsk level can be ncreased by one of the tactcal actons lsted n (10.6). Ths wll also change E[CR[]] and ARAC[], and we should not assume that the changes wll be lnearly related to the change n RAC[]. Note that a change n the rsk portfolo wll also change RBC. For example, let us assume that component s an underwrtng component. Thus the base case defnes a certan level of pure premum PP 0 [] (premum mnus all CARepaper\paperFN9.doc 74

75 external (commssons, brokerage fees, etc.) and nternal expenses). The expected result E[CR 0 []] s smply the expected underwrtng return (ncludng rsk-free nvestment ncome). Ths relates to the expected underwrtng economc margn EUEM 0 [] as a percent of pure premum as follows. (10.11) Expected underwrtng economc margn EUEM 0 [] = E[CR 0 []] / PP 0 [] (defnton) Lkewse we can defne a target prcng margn, or target underwrtng economc margn, TUEM 0 [], and then relate t to the target result, TCR 0 [], for component. Thus t s clear that, gven any assumed rsk portfolo, any desgnaton of an overall target RoE and any allocaton of RAC can be translated nto a target company return and thus nto unque target economc underwrtng margns for each rsk component. And vce versa, any set of target economc underwrtng margns for each rsk component for an assumed rsk portfolo translates back nto an overall RoE and exactly one RAC allocaton. It should also be clear that for an underwrtng component, as RAC[] ncreases by ncreasng the volume and type of busness beng wrtten, E[CR[]] cannot ncrease ndefntely due to market constrants. At some pont, E[CR[]] / ARAC[] reaches a maxmum and begns to decrease. Thnkng about ths for a mnute should convnce you, because the fact that volume s ncreasng n a compettve market means that underwrtng standards are decreasng, so that lower proft busness s beng assumed. Please note that the pure premum PP[] at whch E[CR[]] / ARAC[] CARepaper\paperFN9.doc 75

76 reaches a maxmum for the th subportfolo may not be the pure premum at whch the overall portfolo return s maxmzed. Let us deal wth the second possblty for component, where ts expected return dvded by allocated RAC s less than the overall expected RoRAC, also n case 1 where overall RAC 0 < RBC 0. Ths s the knd of rsk assumpton we would lke to have less of. (10.12) Case 1: RAC 0 < RBC 0 and E[CR 0 []] / ARAC 0 [] < E[RoRAC 0 ] Then decrease RAC[] by decreasng the rsk level of the th component. The rsk level can be decreased by one of the tactcal actons lsted n (10.4). Agan, as noted above, ths wll also change E[CR[]], ARAC[] and RBC. We want to feed some of the ARAC for component nto components where (10.10) holds, thereby ncreasng the overall expected RoRAC. As noted above, t should also be clear that for an underwrtng component, as RAC[] decreases by decreasng the volume and type of busness beng wrtten, E[CR[]] / ARAC[] wll tend to ncrease, because the fact that volume s decreasng n a compettve market usually means that underwrtng standards are ncreasng. Lower proft busness s beng dscarded, so that for the remanng portfolo, the expected underwrtng economc margn n (10.11), EUEM[] = E[CR[]] / PP[], ncreases. Agan, ths cannot ncrease ndefntely. As volume shrnks, eventually the rato of underwrtng expenses to pure premum wll grow enough to decrease EUEM[]. Also, CARepaper\paperFN9.doc 76

77 as volume shrnks, at some pont the company begns to lose underwrtng expertse because t s losng experenced underwrters. Ths wll also tend to decrease EUEM[]. We may thnk of ths problem as one of constraned optmzaton. Defne the functon f as follows. (10.13) f(x[1],..., X[n]) = Σ E[ R[] X[] ] where X[] = RAC[] and R[] = CR[] The problem can then be stated as follows. (10.14) Maxmze f wth respect to {X[1],..., X[n]} such that Σ A[] = RBC {X[1],..., X[n]} where A[] = ARAC[] {X[1],..., X[n]} Standard technques can be used to solve ths problem. But please remember that CR[] s not a lnear functon of RAC[]. In our age of fast desktop PCs, we can bash ths out n a reasonable length of tme. The optmzaton for Case 2, where overall RAC 0 > RBC 0, s smlar. Let us assume that management has no wsh to ncrease RBC, but nstead wshes to decrease RAC. Ths can be solved as was Case 1. Clearly, the soluton wll certanly ental decreasng RAC[] for components where E[CR 0 []] / ARAC 0 [] < E[RoRAC 0 ]. CARepaper\paperFN9.doc 77

78 11. Concluson We have dscussed the reasons why an nsurance company should address rsk and captal ssues n a methodcal manner and we have dscussed some of the problems encountered dong so. We hope that the reader has dscovered some good deas and some procedures for obtanng useful measurements. We hope that the reader found the dscusson of threat scenaros and ther estmaton partcularly useful, snce we beleve that, whatever RAC measure s used, a more accurate measurement of future underwrtng rsk s crucal to proper RAC determnaton. We hope to see less black box modelng of ths problem, and more thought put nto the estmaton of underwrtng parameters especally. We hope that someone devses better ways of modelng and estmatng nvestment rsk over extended tme perods. We hope that the modelng and estmaton of credt rsk s mproved. We hope that the dscusson of the varous possble propertes of captal allocaton methods wll help mprove future dscussons of ths topc. We also hope that the reader also found the dscusson of managng RAC to be nterestng and nformatve. CARepaper\paperFN9.doc 78

79 12. Bblography [0.1] Swss Re, From Rsk To Captal: An Insurance Perspectve, Swss Re publcaton, [1.1] Feldblum, S., NAIC Property/Casualty Insurance Company Rsk-Based Captal Requrements, PCAS LXXXIII, [3.1] Daykn, C.D., Pentkänen, T. and Pesonen, M., Practcal Rsk Theory for Actuares, Chapman and Hall, [3.2] Klugman, S., Paner, H. and Wlmot, G., Loss Models: From Data to Decsons, John Wley & Sons, [3.3] Embrechts, p., Kluppelberg, C. and Mkosch, T., Modellng Extremal Events for Insurance and Fnance, Sprnger-erlag, [3.4] Magnn, J. and Tuttle, D., Eds., Managn Investment Portfolos: A Dynamc Process, Warren, Gorham & Lamont, 1990, p [4.1] Klugman, S., Paner, H. and Wlmot, G., op. ct. [4.2] Embrechts, p., Kluppelberg, C. and Mkosch, T., op. ct. [4.3] Taylor, G., Clam Reservng n Non-Lfe Insurance, North-Holland, [4.4] A.M. Best, Property/Casualty A&E Losses Plunge, But Concerns Reman for Indvdual Companes, BestWeek Specal Report: Property/Casualty Edton, September 21, [6.1] J.P. Morgan/Reuters, RskMetrcs TM Techncal Document, New York, December 1996, free download from [6.2] A comprehensve alphabetcal lstng of publshed value-at-rsk papers can be found at and a lst wth workng papers at [6.3] Franzett, C., alue at Rsk for Longer Tme Horzons, Swss Re nternal document, CARepaper\paperFN9.doc 79

80 [7.1] JP Morgan, CredtMetrcs TM Techncal Document, New York, Aprl 1997, free download from [7.2] Credt Susse Fnancal Products, CredtRsk +, A Credt Rsk Management Framework, 1997, free download from [7.3] KM, Credt Montor, nformaton can be downloaded from [7.4] Wlson, T., Measurng and Managng Credt Portfolo Rsk, techncal document by McKnsey & Co.. [7.5] Wlson, T., Portfolo Credt Rsk (1), Rsk 10, 1997, pp [7.6] Wlson, T., Portfolo Credt Rsk (2), Rsk 10, 1997, pp [7.7] S&P Credt Week, Aprl 15, [8.1] Kreter, F., Bloch, A., Meyer, P., Schmd, E. and Bernegger, S., Catastrophe Portfolo and Captal Needs of a Rensurance Company, Transactons of the 25 th Internatonal Congress of Actuares, Brussels, [9.1] Artzner, P., Delbaen, F., Eber, J-M. and Heath, D., Thnkng Coherently, Rsk 10, [9.2] Artzner, P., Applcaton of Coherent Rsk Measures to Captal Requrements n Insurance, North Amercan Actuaral Journal, vol. 3, [9.3] Chrstoffersen, P., Debold, F and Schuermann, T., Horzon Problems and Extreme Events n Fnancal Rsk Management, Economc Polcy Revew, Federal Reserve Bank of New York, [9.4] Embrechts, P., Resnck, S. and Samorodntsky, G., Extreme alue Theory as a Rsk Management Tool, North Amercan Actuaral Journal, vol. 3, CARepaper\paperFN9.doc 80

81 13. Appendces Appendx alue at Rsk (13.1.1) Goal Model and/or estmate a specfc percentle of the dstrbuton of the change n value of an nvestment portfolo over a gven tme perod. (13.1.2) Assumptons The randomness of the portfolo value stems from the randomness of the values of the component fnancal nstruments, whch s nduced by random rsk factors. Most often t s assumed that the changes of rsk factors are normally dstrbuted. More precsely, the logarthms of the rsk factors are modeled as ndependent dentcally multvarate normally dstrbuted ncrements. Ths assumpton s not questoned here. In the smplest form of ar, the functonal dependences of the nstrument prces upon the rsk factors s lnearzed for the non-lnear nstruments. These are all nstruments wth some optonalty features. CARepaper\paperFN9.doc 81

82 (13.1.3) Defntons Q Y ( t ): = log( Q ( t )) X ( t Q ( t ) Q ( t 1 ) ( t ) = Y ( t ) Y ( t 1) = log( Q ( t Q ( t ) P (Q,t): Instrument prce (value), k = 1,2,...,K k ( Q,t ) ): Rsk = K k = 1 factor, = 1,2,...,J ( t ) = Cov( X ( t ),X P :Total portfolo value k ( t )) ) / Q ( t 1)) We assume a portfolo of K nstruments whch are exposed to J rsk factors. Rsk factors may be nterest rates, stock ndces, foregn exchange rates and commodty prces. Some defntons above have been ntroduced for the sake of smplfcaton. The lnearzaton yelds the followng. (13.1.4) Lnearzaton P ( Q,t ) = k J = 1 P k Q Q = K J k= 1 = 1 P k Q Q = J K = 1 k = 1 P k Q Q Q Q : = Q J T ( w ) = w X = 1 Q wth X = ( X, X w = ( 1 2 P,...,X J ) T, P K k K k K Q, Q,.., k k = 1 1 = 1 2 k = 1 Q Q 1 2 P Q k J Q J ) T. The X vector descrbes the logarthms of the rsk factor changes and w s the vector of exposures or senstvtes. CARepaper\paperFN9.doc 82

83 The varance and the standard devaton of the portfolo value change can be calculated as follows. (13.1.5) arance and standard devaton ar( ) = ar( w = w T Rw T X ) = w T ar( X ) w = w T Rw where the matrx R s the so-called covarance matrx J R ( t ) =..... J1.. JJ The alue at Rsk s defned as a percentle, most often the 95% or 99% percentle, of the assumed normal dstrbuton. (13.1.6) alue at Rsk, ar, wth respect to percentle γ Pr ob( > ar ) = 1 ª. (13.1.7) ar for the normal dstrbuton a R α 99 % = = α γ σ Note: We have omtted the dervatve wth respect to tme. Ths s defensble on the grounds that ts effect do not show up n the ar calculaton due to the determnstc CARepaper\paperFN9.doc 83

84 nature of tme and because the resultng so-called drft term may be added afterwards. So far we have not specfed the tme nterval, but t s mplct n the rsk factor change. (13.1.8) Rsk factor change X ( t ) = log( Q ( t ) / Q ( t )). 1 Therefore s based on the same nterval as the change n rsk factors. Let us assume t to be one day. Under the so-called OLS condtons, ar based upon one day can be scaled to k tmes one day by multplyng wth the square root of k. Frst, let us defne OLS condtons. (13.1.9) OLS condtons: Equal expectatons: E(Y ( t )) = for all t Equal varances: ar(y ( t )) = for all t, and No seral correlaton: Cov( Y ( t ),Y ( s )) = 0, s t Under these condtons, the followng holds. CARepaper\paperFN9.doc 84

85 ( ) OLS condtons mply: ar(y ( t + k ) Y ( t )) = k ar(y ( t + 1) Y ( t )) and Cov({Y ( t + k ) Y ( t )},{Y ( t + k ) Y ( t )}) = k ar({y ( t + 1 ) Y ( t )},{Y ( t + 1) Y ( t )}). Because all varances and covarances are scaled wth k, the portfolo varance s also scaled by the same factor. It follows mmedately that a alue at Rsk based on one day can be scaled to k tmes one day by multplyng wth the square root of k. ( ) ar for k days ar k Days = ar1 k. Day For determnng the captal adequacy of the company, for plannng and allocaton purposes and for makng rsks comparable wth other rsk sources, e.g., credt and nsurance or underwrtng, we are nterested n a rsk measure that has the followng form. ( ) Rsk Measure Form Prob (alue n one year - Expected value n one year > LaR) = p LaR s a percentle to be found. The man dffculty conssts n estmatng realstcally the dstrbuton of the value of the portfolo n one year gven both exogenous uncertantes, e.g., rsk factor changes, and endogenous actons, CARepaper\paperFN9.doc 85

86 especally the strateges of the portfolo managers. Therefore, more precsely we are lookng for a probablty dstrbuton whch s contngent upon the strategy, call t θ. ( ) Rsk Measure contngent upon strategy θ: Prob (alue n one year - Expected value n one year > LaR θ ) = p CARepaper\paperFN9.doc 86

87 Appendx 13.2 The Margnal and the Euler Allocaton Prncples The determnaton of the overall RAC C s based upon a set of volume measures (premums, labltes, number of shares, bonds, etc.) whch represent the portfolo and upon models for the varous rsk factors (natural catastrophes, man-made catastrophes, nterest rates, etc.). For a gven rsk model, C can be represented as a functon ρ, called a rsk measure, of the volume measures. (13.2.1) RAC as a functon of volume ρ,,..., N ) = ( ) = C ( 1 2 ρ If the volume of unt s slghtly modfed,.e., f s replaced by +, then the outcome of the RAC formula wll also dffer slghtly from C. (13.2.2) Change n RAC as volume changes ρ ( 1, 2,..., 1, +, + 1,..., N ) = C + C The rato C / measures the senstvty of the overall RAC, C, wth respect to the rsks belongng to unt. The hgher ths rato, the stronger the overall RAC depends upon the rsks of unt. Of course, the contrbuton also depends upon the actual volume of the rsks. If the RAC allocated to unt I, C, should depend upon the contrbuton of the unt to the overall rsk, then t s very natural to allocate t n proporton to the senstvty and the volume. CARepaper\paperFN9.doc 87

88 CARepaper\paperFN9.doc 88 (13.2.3) Margnal Allocaton: RAC allocaton n proporton to senstvty and volume = = N C C C C e C C 1.. The frst term and the denomnator are requred for the purpose of normalzaton,.e., n order to make sure that the sum of the allocated RAC s, C, equals the total RAC, C. The margnal prncple s addtve for the case of nfntesmal volume changes (see Euler prncple below). For the case of non nfntesmal changes (e.g., for wth and wthout as defned below) t s only almost addtve as demonstrated for followng example. (13.2.4) Example * * * * * * * 2 1 * C C C C a C C C C a C C C a a a + + = + = + + = + = = + = The rsk measure ρ s homogeneous when the followng condton holds: If all volumes are multpled by the same factor λ, then the overall RAC wll also change by the factor λ. (13.2.5) Homogenety of ρ

89 CARepaper\paperFN9.doc 89 C N = = λ λ ρ λ λ λ ρ ) ( )...,,, ( 2 1 If ρ s homogeneous, then the followng relaton s obtaned by takng the dervatve on both sdes. (13.2.6) ( ) ( ) ( ) ( ) ( ) ( ) C C C U U N N U = = = = = = = = ρ ρ ρ λ λ λ ρ λ ρ λ λ 1 1 The Euler prncple splts the overall RAC nto contrbutons from the ndvdual components. It s therefore natural to allocate the RAC n the same way. (13.2.7) The Euler prncple allocaton ( ) C = ρ The dervatves represent the above senstvty (for the case of nfntesmal varatons) and thus the Euler prncple s a specal case of the margnal prncple. It can therefore be easly nterpreted and t has the nce property that t s already normalzed.

90 CARepaper\paperFN9.doc 90 (13.2.8) Example Let s assume that the rsk measure ρ s defned as a multple of the overall standard devaton. In ths case, the Euler-prncple s equvalent to the covarance prncple: ( ) ( ) ( ) ( ) ( ) ( ) [ ] [ ] ρ σ σ σ ρ σ σ ρ σ σ ρ σ σ ρ σ σ ρ σ σ ρ σ σ ρ σ σ ρ = = + + = + + = + = = X X X k k k k k var, cov :, 2 : 2 :, 2 : 2 :, 2 Homogenety s one of the most mportant propertes to be fulflled by coherent rsk measures as defned by Artzner, Delbean, Eber and Heath [ref. 9.2]. It smply says that f one blows up (or down) all volume measures by the same factor, the overall RAC wll also ncrease (decrease) by the same factor. Apart from captal measures expressed n absolute and not relatve terms, such as mnmum statutory captal requrements for example, all relevant rsk measures used n practce (percentages of volume, standard devaton, percentle, shortfall, etc.) are homogeneous. From ths perspectve, the Euler prncple would be the natural canddate for RAC allocaton. However, f certan RAC components lke the RTL are defned on the bass of external rsk percepton, the Euler prncple wll also determne the contrbuton of a unt to ths component. That s, RAC allocaton would then be at least partly based on external

91 rsk percepton, whch s not necessarly n lne wth the obectves of RAC based busness decson-makng. Of course, the Euler prncple cannot only be appled to the overall RAC, but to any homogeneous rsk measure. It wll therefore have to be consdered as an allocaton canddate even n the case where dfferent rsk measures are used for RAC determnaton and RAC allocaton. (13.2.9) Examples of allocaton va the Euler prncple If the overall RAC (or one of ts components as, e.g., the RTL) s gven as the sum of weghted volumes, then there s no dversfcaton and the allocated RAC of unt equals the standalone RAC of unt. If the overall RAC s gven as a functon of the standard devaton of the overall result dstrbuton and the f the homogenety prncple s fulflled, then the RAC can only be a multple of the standard devaton! In ths case, the Euler prncple corresponds to the covarance prncple as shown above. If the overall RAC s derved from percentles, shortfall rsk or smlar rsk measures, t can be represented as a functon of all or several moments of the overall dstrbuton. Accordng to the Euler prncple, the contrbuton of the ndvdual components can be represented as a lnear combnaton of co-moments. Another specal case of the margnal prncple s the wth and wthout prncple. Here, the overall RAC s evaluated wth and wthout the rsk of unt, and the RAC s CARepaper\paperFN9.doc 91

92 then allocated n proporton to the RAC reductons. Ths can be acheved by settng the volume change equal to - n the above formula for the margnal RAC allocaton. CARepaper\paperFN9.doc 92