Quantifying Uncertainty in Reserve Estimates

Transcription

1 Quantfyng Uncertanty n Reserve Estmates by Za Rehman and Stuart Klugman ABSTRACT Property/casualty reserves are estmates of losses and loss development and as such wll not match the ultmate results. Sources of error nclude model error (the methodology used does not accurately reflect the development process), parameter error (ncorrect model parameters), and process error (future development s random). Ths paper provdes a comprehensve and practcal methodology for quantfyng rsk that ncludes all three sources. The key feature s that varablty s captured by examnng hstorcal changes n ultmate values rather than examnng the underlyng clam dstrbuton. We present the conceptual framework as well as practcal examples. KEYWORDS Reserve uncertanty/varablty, reserve allocaton, multple lnes 30 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 1

2 Quantfyng Uncertanty n Reserve Estmates 1. The varablty problem 1.1. The challenge of reservng The property/casualty busness model reles on the accurate measurement of rsk. Of relevance to ths paper s the measurement of reservng rsk. Accurate actuaral loss reservng s one of the regulatory requrements n measurng solvency. Consequently, one of the most mportant tasks for an actuary s to estmate the proper amount of reserves to be set asde to meet future labltes of current n-force busness. Because the stated reserve s an estmate and not the true number, there s error and t s mportant to measure ths error. Quantfyng the potental error allows for settng ranges around a best estmate, allows for measures of rsk, and can assst n the settng of rsk-based captal requrements. Recent papers such as Hayne (2004) and Shapland (2007) make the mportance of ths ssue clear. In ths paper we propose a method for measurngthetotalrsknvolvednreserveestmates.it s smple to apply and uses data that s almost always avalable from the reserve-settng process. A key feature s that our measure of reserve varablty does not depend on the method used to determne the reserves Lterature revew There are many papers regardng dfferent loss reservng technques, some determnstc, some stochastc. For a comprehensve revew of exstng determnstc methods, the reader s referred to Wser (2001) and Brown and Gottleb (2001). For an excellent overvew of a wde range of stochastc reservng methods n general nsurance, the reader s referred to England and Verrall (2002). Other references of nterest nclude Bornhuetter and Ferguson (1972), Fnger (1976), and Taylor (2000). Whle the tradtonal chan ladder technque provdes only a pont estmate of the total reserve, t has become evdent recently that actuares also need a measure of varablty n loss reservng estmaton. Sound methodologes that quantfy rsks related to the balance sheet are mportant for attractng and retanng captal n the frm. Standard and Poor s ratngs as well as nvestorsareverynterestednthevalueatrsk (VaR) measurements. The NAIC and state regulators are nterested n montorng the reserve varablty n the form of reserve ranges. Varablty and sometmes the entre dstrbuton of the loss reserve estmate are mportant for rsk management purposes. Questons such as what s the 95th percentle of losses or the cost of loss portfolo transfer are mportant n managng and assessng rsk. These questons can be properly addressed wth a thorough analyss of the varablty of the reserve estmate. Over the last 30 years many researchers have made sgnfcant contrbutons to the study of the varablty of reservng methods. The CAS workng party paper (2005) presents a comprehensve revew that brngs all of the mportant hstorcal research together. A representatve, but not exhaustve, lst follows. Taylor and Ashe (1983) ntroduced the second moment of estmates of outstandng clams. Hayne (1985) provded an estmate of statstcal varaton n development factor methods when a lognormal dstrbuton s assumed for these factors. Verrall (1991) derved unbased estmates of total outstandng clams as well as the standard errors of these estmates. Mack (1993) used a dstrbuton-free formula to calculate the standard errors of chan ladder reserve estmates. England and Verrall (2002) presented analytc and bootstrap estmates of predcton errors n clams reservng. De Alba (2002) gave a Bayesan approach to obtan a predctve dstrbuton of the total reserves. Taylor and McGure (2004) appled general lnear model technques to obtan an alternatve method n cases where the chan ladder method performs poorly. Verrall (2004) mplemented Bayesan VOLUME 4/ISSUE 1 CASUALTY ACTUARIAL SOCIETY 31

3 Varance Advancng the Scence of Rsk models wthn the framework of generalzed lnear models that led to posteror predctve dstrbutons of quanttes of nterest. The CAS workng party (2005) ultmately concluded, there s no clear preferred method wthn the actuaral communty. Actuares need to select one or several methods that are consdered approprate for the specfc stuaton. When t comes to the fnal decson, judgment stll overrules The rsk measurement problem The CAS workng party report (2005) notes that the sources of uncertanty n the reserve estmate come from three types of rsk: process, parameter, and model. Model rsk s the uncertanty n the choce of model. Parameter rsk s the uncertanty n the estmates of the parameters. Process rsk s the uncertanty n the observatons gven the model and ts parameters. Shapland (2007, p. 124) hghlghts the mportance of all the rsks: Returnng to the earler defnton of loss labltes:::all three types or rsk:::should be ncluded as part of the calculated expected value. Alternatvely, some or all of these types of rsk could be ncluded n a rsk margn as defned under ASOP No. 36. These three rsks are ntertwned and thus hard to separate. In partcular, process and parameter rsk are often calculated as f the model were correct. In ths paper, the three rsks wll not be separately measured nor drectly treated. 1 Rather, we capture the total rsk from all three sources underlyng the reserve estmate. The reservng model (or ultmate loss selecton f no specfc method s used) wll be consdered fxed and any errors measured wll be a consequence of that choce. 1 The model presented wll capture the parameter and model rsk n the actuary s estmate but wll not measure the parameter uncertanty due to ts own estmate. From a rsk management perspectve, ths s approprate. Now that the reserves have been establshed, what s the potental error that may result, gven the current reserve revew? 1.4. A summary of our approach Most approaches to rsk measurement rely on the statstcal propertes of the data as reflected by the model selected for calculatng the reserve. Those approaches attempt to capture the underlyng dstrbuton of losses. In ths settng, parameter estmaton error can be estmated usng statstcal measures such as Fsher nformaton. We choose to look at the reserves (as reflected n the estmated ultmate losses) themselves as they evolve over tme. Ths provdes a way to reflect all the sources of error. Each reserve set n the past s an estmate of ts dstrbuton and thus ts errors can be estmated from the hstorcal errors made n the estmatons. Because the ultmates wll converge to the true value, the errors made along the way reflect all sources of error. Our methodology wll be ntroduced wthn a stable context. In partcular, ntally we assume that the dstrbuton underlyng the loss development process s constant over tme and that we are workng wth ndcated reserves developed under a consstent methodology. However, t s mportant to note that, under these assumptons, the ranges produced do not nclude the possblty of a change n the dstrbuton, such as nflaton movng to a new level not n the avalable hstory or sharp ncreases n the deductbles or retentons wrtten over or ceded. After workng through ths stuaton we wll dscuss modfcatons that can allow our model to be appled n more general settngs. Note that we model ndcated reserves nstead of held reserves. Ths s because ndcated reserves do not have management or other ad hoc adjustments. Therefore, ndcated reserves are more stable and thus easer to model. What s nterestng about held reserves s where they fall wthn the probablty dstrbuton of the un- 32 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 1

4 Quantfyng Uncertanty n Reserve Estmates known true reserve. Ths can provde an ndcaton of the degree to whch held reserves are conservatve or aggressve. Because the method presented here s free of the choce of the reservng method used, t s not necessary to even have a specfc method. The only requrement s a hstory of ultmate loss selectons. Thus we rely on the actual error hstory of the reservng department. We make a theoretcal and pragmatc case for the lognormal dstrbuton for the errors n aggregate reserves, lne or total. The focus on the aggregate dstrbuton also removes the need to choose ndvdual sze of loss dstrbutons. Each of the followng sectons wll take one step through the development of the rsk measure. An example wll be followed throughout to llustrate the formulas. 2. Reservng process and data The reserve revew process generates reserves based on raw data analyss. The ndcated ultmate losses are selected by lne usng perhaps several methods as well as judgment. These lne ultmates are then added to yeld the total ndcated reserve. Management adjustments called margns may be appled to the total reserve. These margns may then be allocated by lne and by accdent year to the ndcated ultmate losses and reported n Schedule P Part 2. As noted earler, our method does not work wth these reserves. There are three ssues of nterest relatng to measurng reserve *varablty: (1) The dstrbuton of the true (but unknown) ultmate losses by lne and n aggregate. Ths shows the volatlty and the bas n the actuaral selectons. (2) The held reserves that are reported n Schedule P Part 2 as a percentle on the dstrbuton of reserves. (3) A procedure to allocate the margn by lne and accdent year such that ultmate losses for all years are at a constant percentle. It s nstructve to understand several aspects of ndcated reserves. ² Indcated reserves are the actuary s best estmates based on the data and exclude management adjustments. Therefore, errors are due only to actuaral selectons, methods, or randomness n the data. ² Data trangles used for reserve revews can be quarterly or annual. Generally, most companes lke to have consstency n the ndcated revews because t s easer to update spreadsheets for each revew f they are the same sze. Also, most companes lke to track development to ultmate and prefer complete trangles f data s avalable. ² Reserve revews are done on a net and or gross bass and the underlyng trangles are based on the relevant data. They are defntely conducted annually to report Annual Statement reserves but many companes do them quarterly. ² Reservng actuares refer to lnes of busness as segments, as they can be custom-defned by the actuary for reserve revews. These can be dfferent than the usual lnes of busness such as those defned n Schedule P. For consstency of notaton n ths paper, we wll refer to reserve segments as lnes of busness. The methodology wll be the same n both cases. Generally the DCC (Defense and Cost Contanment) reserve revew (ndcated reserves) s done usng data where they are ether a part of the loss trangle (loss+dcc) or are treated separately (DCC only). In the frst case we treat DCC as part of the losses, and n the latter case as another data segment (f broken out separately). In ths paper, loss shall mean whatever appears n the analyss beng evaluated. The varablty for ULAE reserves s outsde the scope of the paper and wll not be dscussed. For most companes ULAE reserves are a relatvelysmallpartofthetotalreserves. VOLUME 4/ISSUE 1 CASUALTY ACTUARIAL SOCIETY 33

5 Varance Advancng the Scence of Rsk 2.1. Scope of model The approach presented n ths paper s generc and apples to any type of trangle, such as ² Pad or ncurred ² Count or Severty ² Accdent year, polcy year, or report year ² Quarterly or annual. Each of these trangles eventually leads to the approprate ultmate losses. Dependng on the choces made above and the selected reserve methodology, the dstrbuton of errors wll dffer. For example, f only the pad loss development method s used and we track ts hstory of ultmate losses over tme, then the resultng reserve dstrbuton wll pertan to the pad loss development method. If several methods are used and the actuary fnally selects reserves based on several ndcatons (typcally ths s the case), then the trangle hstory of fnal selected ultmate losses wll provde the dstrbuton of the selected reserves. An nterestng pont s that the method can even be appled to the raw data. Ths s equvalent to treatng the data as the selected ultmate loss. In ths case, the resultng dstrbuton wll pertan to the data tself (pad or ncurred losses, etc.). Ths essentally creates a new reservng method. However, the purpose of ths paper s not to promote a new way of calculatng reserves, but to develop a method for determnng the dstrbuton of the ndcated reserves resultng from actuaral selectons Reasonable estmates A standard assumpton that s n lne wth Actuaral Standards of Practce (ASOP) on reserves s that the reserve ranges are set around reasonable estmates. Ths s partly acheved by usng ndcated ultmate reserves nstead of held reserves, as these do not have management adjustments. Cases where the ndcated ultmate losses are unreasonable are outsde the scope of ths paper. Ths does not necessarly mean that the model wll not apply, but rather that the authors have not gven consderaton for such cases n ths paper. 3. A model for errors 3.1. Measurng the error from the data The notaton wll be llustrated wth an example that wll be carred through the paper. Suppose for a partcular lne we beleve that losses are fully developed after 10 years. There have been 12 reserve revews completed and n each year an ultmate loss has been estmated. For notaton, let U k the estmated ultmate loss as of calendar year k for accdent year. The results for an example block of busness are n Table 1. Note that the avalable data has and k range from 1 through 12 but, for example, the accdent year 1 losses were fully developed by calendar year 10. For example, the value 42,894 s the ndcated ultmate loss estmated at the end of calendar year 10 for losses ncurred n accdent year 6. We are nterested n the errors made n the estmates of the ultmate losses. Some of those errors can be determned from Table 1. We wll use the logarthm of the rato for the errors, for reasons to be explaned later. For example, the ultmate value for accdent year 3 was estmated at the end of calendar year 11 (development year 8) to be 85,626. A year later, the actual value of 85,650 was known. The error s ln(85,650=85,626) = 0: We cannot calculate the error for later accdent years because they have not yet been fully developed. The errors we care about are e k =ln(u+9 =U k ) where +9>k. The numerator s the fully developed ultmate loss and the denomnator s the estmate as of calendar year k. For our example we are currently at k = 12 and so are concerned 34 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 1

6 Quantfyng Uncertanty n Reserve Estmates Table 1. Indcated ultmate losses from 12 calendar-year (CY) valuatons AY()=CY(k) , , ,010 98,001 96,280 95,579 95,176 95,161 95,150 95, , , , , , , , , , , ,092 94,318 89,032 86,552 85,584 85,532 85,557 85,655 85,626 85, ,441 52,585 52,136 51,375 51,501 51,799 51,870 51,914 51, ,738 30,670 32,948 33,986 34,363 34,329 34,467 34, ,134 37,035 42,981 42,688 42,894 43,052 43, ,695 51,849 57,143 57,817 58,200 58, ,397 67,688 74,995 75,793 76, ,537 94,281 98,453 98, , , , , , ,224 Table 2. Year-to-year error values, e j AY()nDY(d) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : wth the errors made n estmates from accdent year 4 onward. In order to gather more nformaton about errors, begn wth an error that s not mmedately =U j )whereaval- useful. Consder e j =ln(u j+1 able. Ths represents the error realzed as the estmated ultmate value s updated one calendar year later. Here s one pragmatc reason for usng logarthms the errors are addtve. In fact, e k =ln(u +9 =ln à U k+1 U k +8 k = g=0 +8 = e j : j=k =U k ) U k+2 U k+1! U+9 U +8 ln(u k+g+1 =U k+g ) In ths notaton, the development year of the denomnator value s d = j + 1. The advantage of ths approach s that the avalable data n our example provdes many estmated values, as presentedntable2. There s another nterpretaton of these errors. Regardless of the reservng method, a factor representng the ultmate development can be nferred. Suppose we are lookng at accdent year and calendar year j. The factor s u j = Uj =Lj where L j s the pad loss for that accdent year at the end of year j. One year later the factor s u j+1 = U j+1 =L j+1. These represent age to ultmate for two dfferent development years. The rato u j =uj+1 represents how losses were expected to develop, whle L j+1 =L j s how they actually developed. The rato of actual to expected s U j+1 =U j, whch s the error measurement we have been usng. VOLUME 4/ISSUE 1 CASUALTY ACTUARIAL SOCIETY 35

7 Varance Advancng the Scence of Rsk Now that the key data values have been calculated, t s tme to construct a model. The model We propose that the error random varables have the normal dstrbuton, and n partcular, e j» N(¹ j +1,¾ 2 j +1 ): Note that the mean and varance are constant for a gven development year. The rest of ths secton s devoted to justfyng these assumptons. It should be noted that justfyng the assumptons through data analyss s not suffcent. If ths method s to be useful regardless of the loss reservng method used, the justfcaton must be based on our belefs about the loss development and reservng processes and not any partcular models or emprcal evdence. No ndependence assumptons are made and the covarance measurement wll be presented formally as part of the model Normal dstrbuton After the ultmate loss s estmated at a partcular tme, what factors wll cause t to change when t s re-estmated one year later? Day by day durng that year a varety of events may take place: 2 ² Economc forces such as nflaton and changes n the legal envronment wll alter the amounts pad on open clams or those newly reported. ² The rate at whch clams develop may change. ² Purely random events may affect ndvdual open clams. ² The actuary s opnon on IBNR may change, dependng on the adequacy of case reserves. These factors wll tend to act proportonally on the current estmate of the ultmate loss. Because there are many such factors happenng many tmes n the course of a year, t s reasonable to assume we are lookng at the product of a varety of random varables, most wth values near 1. The Central Lmt Theorem ndcates that the result s a lognormal dstrbuton and thus measurng the error n the logarthm wll produce values wth a normal dstrbuton. We recognze that ths reasonng s not accepted by all actuares but we feel t s a reasonable startng pont Normal approxmaton The above arguments relatng to the central lmt theorem apply to a fcttous accdent year wth nfnte clams. Ths accdent year need not be evaluated nfntely many tmes but the changes at each valuaton should be drven by nfnte reasons underlyng nfnte clams. In practce, a fnte subset of ths hypothetcal accdent year s avalable. The reasons causng the change n the ultmate loss are fnte and not nfnte and thus the dstrbuton s approxmately normal. 3 The approxmaton can be mproved by ncreasng the number of reasons drvng the change n ultmate losses. Ths can be done n the followng ways: ² Increase the valuaton tme (for example, once a year rather than four tmes a year). ² Increase the clam count (large volume lne). The clam count requred for a good normal approxmaton n turn depends on the skewness of the underlyng sze of loss dstrbuton. As a practcal matter, the model wll work well even for a modest clam count. Reserve valuatons conducted less frequently (for example, once a year rather than four tmes a year) wll allow greater reasons for changes and thus help wth the normal approxmaton. The tradeoff s the loss of accuracy n ultmate loss estmaton and consequental bas. Also note 2 At ths pont n the paper we focus on lognormalty as a consequence of unchangng development processes. We relax ths assumpton later n the paper. 3 Techncally, the dstrbuton remans approxmately normal even for the nfnte clam accdent year, but the dstrbuton s closer to normal than n the fnte case. 36 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 1

8 Quantfyng Uncertanty n Reserve Estmates that both the parameters of the dstrbuton and ts shape (to whch extent t s lognormal) wll change dependng on the frequency of reserve revew. A subtle pont concerns the later development ntervals. In these ntervals, the changes are usually drven by a handful of clams (few reasons) and thus volate the normalty assumpton. However, at those valuatons the aggregate errors are often close to zero wth small volatlty (constants) and thus departure from normalty has lttle mpact on the total dstrbuton for the entre accdent year Mean At frst t may seem that the mean should be zero. There are two reasons that s not so. Frst, consder the expected revsed ultmate loss gven the current estmate: E(U j+1 =U j )=e¹ j +1 +¾ 2 j +1 =2 : For the reserve estmate to be unbased, t s necessary that ¹ j +1 = 0:5¾ 2 j +1 and not zero. In addton, t s possble that the reservng method s based. Ths may be a property of the method selected and may even be a delberate attempt by the reservng actuary to adjust the reserves based on knowledge that s outsde the data. Havng the mean be constant from one accdent year to the next s a consequence of the assumptons that were made at the begnnng. That s, there s no dstrbutonal change n the underlyng development process or reservng method. Havng the mean depend on the development year seems reasonable. As the development year ncreases, any systematc bas s lkely to be reduced n the expectaton that the ultmate value wll not be much dfferent from the current value. In addton, we expect the varance of the errors to decrease, and f the ultmate loss estmates are unbased, the means wll then also decrease (n absolute value) Complete trangle of errors For at least one accdent year we need the losses to be fully developed. Even better would be to have a few fully developed years so there would be more data avalable for computng covarances and varances. There s a tradeoff: older fully developed accdent years that are not part of the n-force book may be less predctve and may not reflect the current prevalng busness envronment. The opposte concern s when there are not enough accdent years avalable to obtan fully developed losses. Ths can happen for relatvely new companes or lnes of busness. Note that tal errors generally result from estmaton errors of pendng court cases or smply absence of data due to a new lne. Snce we are dealng wth a year n aggregate, these errors are usually close to zero and therefore contrbute less to the varance than earler development ntervals. 4. Parameter estmaton Return to the contnung example and recall Table 2. The numbers n ths table are percentage changes (errors) of the estmated ultmate losses andhavebeenassumedtohavebeendrawnfrom sx normal dstrbutons. The upper half of the trangle s fxed and known and our goal s to obtan the normal means and varances for the yet-to-be observed errors n the lower rght part of the table Estmaton of the mean We choose to calculate an ntal estmate of the mean by calculatng the sample mean, ˆ¹ d = P 12 d =1 e(+d 1) 12 d where d s the development year, the sum s taken over all accdent years for whch errors are, VOLUME 4/ISSUE 1 CASUALTY ACTUARIAL SOCIETY 37

9 Varance Advancng the Scence of Rsk Table 3. Estmated means by development year DY(d) mu-hat : :0002 mu-hat : :0002 (Selected) Table 4. Standard devatons by development year DY(d) sd-hat avalable. 4 The sample mean s also the maxmum lkelhood estmate of the true mean and works best f the data s stable. If the dataset s not stable, then other estmates such as those nvolvng tme seres analyss can be used. The results for our example are n Table 3. However, there are reasons why the sample mean may not be the approprate choce. The results may be based due to model rsk. The estmate of the mean affects the estmate of the expected ultmate loss. For example, f ¹< ¾ 2 =2thentheestmated ultmate loss wll be less than the ndcated ultmate showng redundancy n resultng reserves. The opposte s true f the nequalty s reversed. Thus any value of mean that s dfferent than ¾ 2 =2 should be justfed. The estmaton of the mean can effectvely result n takng a poston that the actuaral estmate s based. Ths s not trval and there should be careful analyss before makng that asserton. We dscuss the mean selecton n greater detal under dstrbuton revews later n the paper Estmaton of the varance The varance parameter measures the rsk hstorcally faced by the book n force. The varance estmate s the usual unbased estmate. The 4 Analternatvestouseaweghtedaveragewheretheweghts are the ndcated ultmate values for that accdent year. Ths allows more weght to be placed on those accdent years n whch there s more data. equaton s P 12 d ˆ¾ d 2 =1 = [e(+d 1) ˆ¹ d ] 2 : 11 d Note that the estmated mean must be the sample mean and the above s not the maxmum lkelhood estmate. For our data the estmated standard devatons by development year are gven n Table 4. As expected, for the most part, the standard devatons decrease by lag Correlatons For a gven accdent year, the errors for one development year may be correlated wth those from other development years. Any such correlatons can be estmated. A formula for the covarance s P 12 d ˆ¾ d,d 0 = [e (+d 1) ˆ¹ d ][e (+d0 1) ˆ¹ d 0] 11 d where d>d 0 represent two dfferent development years and each sample mean s based only on the frst 10 d observatons. ThematrxofcovarancessgvennTable5. The total varance for any gven accdent year can now be calculated from Table 5. For example, the varance for accdent year 12 wll be the sum of the values n the table (shown as n Table 6). The correlaton matrx underlyng the above covarances should be tested for statstcal sgnfcance. Ths s outsde the scope of ths paper. 38 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 1

10 Quantfyng Uncertanty n Reserve Estmates Table 5. Covarances d The error dstrbuton for a gven accdent year Now that we have a model for the errors from one calendar year to the next, we need to return to the total error. Recall that +8 e k = e j : j=k From the assumptons made prevously, ths error has a normal dstrbuton and ts mean depends only on the latest development year, k +1. e k» N(¹ k +1,¾ 2 k +1 ) ¹ k +1 = +8 ¹ j +1 j=k +8 ¾k +1 2 = +8 ¾ j j=k j=k+1 j 0 =k j ¾ j +1,j 0 +1 : These quanttes can be estmated by addng the respectve mean, varance, and covarance estmates. For our example, we now have estmates of the dstrbuton of the error n the ultmate loss as estmated from the data avalable at the end of calendar year 12. They are gven n Table The error dstrbuton for all accdent years combned Whlewehavebeenusnglogarthmstomeasure the error, when all s done we are nterested n the ultmate losses themselves. To make the formulas easer to follow, rather than allow ar- Table 6. Means and standard devatons by accdent year (AY) AY mean sd 4 0: : btrary values, we wll follow the example and assume we are at the end of calendar year 12 and losses are fully developed after 10 years. In partcular, we care about U = U U : Recall from the notaton that the terms on the rght-hand sde represent the fully developed losses at a tme n the future when the ultmate results are known. Rewrte ths expresson as U = U4 12 exp(e12 4 )+ + U12 exp(e12 ) r = = V 12 =4 r exp(e 12 ) U 12 U U12 12 V = U U : Here V s the estmated ultmate loss as of calendar year 12 for all open years and the weghts are the relatve proportons n each accdent year. Ths ndcates that the ultmate loss s a weghted VOLUME 4/ISSUE 1 CASUALTY ACTUARIAL SOCIETY 39

11 Varance Advancng the Scence of Rsk average of lognormal random varables. Ths can be panful to work wth (though not hard to smulate). Because the error random varables are usually close to zero and wll vary about ther mean, consder the followng Taylor seres approxmatons. 2 3 =ln U 12 V =ln 4 r exp(e ¼ ln 4 12 =4 0 =ln@1+ ¼ 12 =4 r e 12 : =4 3 r (1 + e =4 ) r e 12 1 A ) The approxmate log-rato has a normal dstrbuton and thus U has an approxmate lognormal dstrbuton. The moments of the normal dstrbuton are: μ E ln U V 0 = = 12 =4 12 =4 = ¹: 0 μ Var ln U ¼ V = r e 12 r ¹ =4 12 r 2 ¾2 13 =4 = ¾ 2 : 1 A r e 12 1 A For the example, V = 760,808, ¹ =0:01927, ¾ 2 =0:01123, and then the expected ultmate loss s 779,978 and the standard devaton s 82, Extenson to multple lnes of busness Suppose there are two lnes of busness. Each can be analyzed separately usng the method prevously outlned. However, t s lkely that the results for the lnes are not ndependent. One approach would be to model the correlaton structure between error values from the two lnes. The problem wth that approach s that there may not be correspondng cells to match and also the number of parameters may become prohbtve. 5 An alternatve s to combne the data from the two lnes nto a sngle trangle and analyze t usng the methods of ths paper. When fnshed, there wll be a dstrbuton for each lne separately and for the combned lnes. Usually, only the total reserve s of mportance and thus only the combned results are needed. The ndvdual lne results wll be useful for nternal analyses and also f t s desred to allocate the reserves back to the lnes. Ths method assumes that the relatve szes of the lnes has been consstent over tme. Adjustments to the hstorcal trangles can be made f ths s not the case, as dscussed later. To llustrate ths dea, we add a second lne of busness. The same analyss gves: V = 244,537, ¹ = 0:30759, ¾ 2 =0:008933, and then the expected ultmate loss s 180,593 and the standard devaton s 17,107. Combnng the two lnes creates a sngle table wth the totals from each. An analyss of these tables produces V = 1,005,376, ¹ = 0:02674, ¾ 2 =0:009582, and then the expected ultmate loss s 983,520 and the standard devaton s 96, Allocaton of ultmate losses An llustratve allocaton model s presented here but other models are also possble. From the prevous examples there s an nterestng stuaton. If the two lnes were ndependent, the standard devaton of the total would be q 82, ,107 2 = 84,638 5 Combnng dstrbutons also requres the multvarate normalty assumpton. Ths s hard to test but s often made n practce. 40 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 1

12 Quantfyng Uncertanty n Reserve Estmates whch s less than the standard devaton from the combned lnes, whch was 96,506. There s thus an mpled postve correlaton between the lnes and when settng reserves and then allocatngthemtothelnes,somethngneedstobe done. Suppose we set reserves wth a margn for conservatsm based on the results of the prevous secton. For example, suppose we set ultmate losses to be at the 95th percentle. That s, ³ 1,005,376exp 0: :645 p 0: = 1,149,833 where the mean and standard devaton are for the correspondng normal dstrbuton. We also want to set ultmate losses for each lne of busness such that they add to 1,149,833. However, ths cannot be acheved by settng each lne at ts 95th percentle. One possble approach s to set each lne at the same percentle, usng the percentle that makes the sum work out. It turns out that f each lne s set at the 96.28th percentle, the ultmate estmates wll be 937,025 and 212, Model extenson to practcal settngs To ths pont we have assumed constant dstrbutons and reservng methods. In practce these are not always present, so some extensons are dscussed here Dstrbuton revews Once the reservng actuary has completed the reserve revew the dstrbuton revew should follow. The actuary wll remember the consderatons for selectng the ultmate losses. These nclude data consderatons, coverage changes, mx changes, etc. All of these can now be factored n the bas (mean parameter) estmaton of resultng reserve dstrbuton. The dstrbuton revew allows the actuary to consder the possblty of estmaton bas n the current reserve estmate. For example, f the hstorcal errors of the selected ultmate losses for a lne are postve and the actuary has not changed the selecton approach, then the current estmate wll lkely have a postve error (too low an estmate). The reserve dagnostcs are partcularly mportant n evaluatng such errors, as the past s not always ndcatve of the future. For example, a sgn of case reserve weakenng should lead to a hgher IBNR, all else beng equal. These mechansms of montorng errors allow early warnng sgns for future potental reserve defcences. Fnally, note that changes n condtons do not nvaldate the lognormal property. However, changes n condtons wll change the lognormal parameters, thereby ncreasng the degree of dffculty for projecton accuracy Changng nternal/external condtons Whle the dstrbuton revews and the resultng mean parameter wll account for changes n reserve adequacy, the volatlty parameter reflects hstorcal data and s therefore slow to respond to changes n the errors. Thus, f the reservng methods change abruptly or the reservng actuary s replaced, the volatlty wll change slowly as the data emerges. Such cases can be handled n the followng two ways: ² In many cases t s wse to let data lead the way as t may be preemptve to conclude about the volatlty of the new process. Ths s especally relevant n lght of the fact that processes themselves usually change slowly over tme (such as experence of reservng actuary, etc.). ² In some cases, the change s drven by an abrupt decson, such as outsourcng the reserve revew to a consultant or change n treaty lmts. In that case, the hstorcal error data wll VOLUME 4/ISSUE 1 CASUALTY ACTUARIAL SOCIETY 41

13 Varance Advancng the Scence of Rsk have to be restated wth the new process. Ths nvolves recalculaton of the past reserve estmates adjusted for the new data and usng the new methodology. To be explct, the volatlty measurement n the model reles on two man assumptons: (1) The data s consstent n terms of reserve model/process and retentons. (2) Future volatlty n ultmate loss selectons can be measured from hstorcal data. Note that we are askng the future volatlty n loss selectons to be smlar to the past and not the consstency n nterest rates, etc Booked reserve The model was used on ndcated reserve data for consstency reasons snce management adjustments can often lead to consstency ssues. If the held reserve data s relatvely consstent, then the model wll perform well wth that data (Schedule P Part 2), as well Lognormal assumpton We present here two dscussons: Dscusson 1: The ultmate losses are hypotheszed to be log-normally dstrbuted and therefore the sums of such lognormal dstrbutons are not exactly lognormal. We frst note that at an accdent year level we do not add lognormal random varables. In other words, we study the gven data that has been obtaned from an underlyng reserve revew process and not artfcally created by an addton process. These two thngs are not the same, as the belef n the underlyng process s the key to descrbng the correct modelng dstrbuton. In nstances where we add lognormal random varables (such as Secton 6) we show mathematcally that f the cumulatve accdent year error e s relatvely small, then the sums of lognormal random varables wll be approxmately lognormal. Note also that the weghted average s approxmately lognormal, so addng a nonlognormal accdent year ultmate loss random varable wth a small weght wth a lognormal random varable wth a large weght wll stll produce a lognormal random varable. The weght s based on selected ultmate loss. Dscusson 2: Why not use any other dstrbuton besdes the lognormal? Besde the theoretcal argument gven for the use of lognormal, we suggest performng an emprcal test on the data. In the end, the model does not depend on the lognormal assumpton and any other dstrbuton can be used, but we beleve that t s the best dstrbuton that descrbes the underlyng reserve process. Addtonally, the lognormal has the advantage that the mathematcal calculatons are tractable. Wth other dstrbutons the calculatons are much harder. The Q-Q plot s a smple way to test for normalty, the error trangle can be standardzed usng the mean and varance of each column of errors and the graph nspected for N(0,1) Changes n volume In Secton 7 we added the ultmate loss data from dfferent lnes and appled the model to the total dataset. Ths approach assumes relatve consstency n volume by lne Analytcal approach An alternatve s to measure the total dstrbuton from the parts. Suppose we know the ndvdual lne lognormal dstrbutons. Let V m = Total (for all accdent years) selected ultmate loss for lne m. We showed earler that for a gven lne, e m =ln U m V m» N(¹ m,¾ 2 m ): 42 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 1

14 Quantfyng Uncertanty n Reserve Estmates We now combne ndvdual lnes, =M U = w m = U m = V m expe m = V V m P V ; V = m ln U V =ln =ln à 1+ à ln U V» N V m w m expe m ¼ ln w m ¹ m,! w m e m ¼ k=1 w m (1 + e m ) w m e m w m expe m! k=m w m w k cov(e m,e k ) : The above used Taylor seres approxmaton of the exp and ln operator. To measure the covarance term n the above expresson, we need addtonal notaton. Let j = Development nterval: Due to the lognormal assumpton of the total ultmate loss dstrbuton, we can vew the lne error as the sum of errors by development nterval, e m = j =1 j=j e mj : Thus, 0 j=j l=j cov(e m,e k )=cov@ e mj, j=j l=j j=1 e kl l=1 = cov(e mj,e kl ): j=1 l=1 1 A Smulaton approach The analytcal approach uses the frst-order Taylor seres approxmaton to ensure that the total reserve dstrbuton for all lnes combned s lognormal. Ths means that we have used the Taylor approxmaton twce at a lne level and then at the total reserve level. To avod approxmatng twce, we can smulate the lne dstrbutons usng the covarance shown below: cov(u m,u n ) 0 =cov@ 0 =cov@ 0 =cov@v + = k=m k=1 k=m V m expe m, k=1 k=m V m (1 + e m ), k=1 k=m V m e m,v + V m V k cov(e m,e k ): 1 V k exp e A k 1 V k (1 + e k ) A k=1 1 V k e A k The above used the Taylor approxmaton of the exp operator. 10. Applcatons We now present a few applcatons of the prevous results Loss estmaton method The chan ladder estmate s not based under loss development factor ndependence assumpton of Mack (1993). If the development factors are correlated, then the model provdes a way to ncorporate covarance of loss development factors nto the chan ladder estmate. Usng the technque presented n ths paper (ncludng the varance covarance matrx of a trangle) the expected ultmate loss for an accdent year s à E(U) = U o exp ¹ + ¾2 2!, where U = Ultmate loss, U o = Incurred or pad loss, ¾ 2 = sum of varance covarance matrx (the number of terms depend on the age of the accdent year) Clam commutatons Clam commutatons nvolve transfer of reserves from a cedng carrer to an acqurng carrer. The rsk to the acqurng company s that the VOLUME 4/ISSUE 1 CASUALTY ACTUARIAL SOCIETY 43

15 Varance Advancng the Scence of Rsk ndcated reserves may not be suffcent to pay all clams. One complcaton s that the agreed-to reserve value may not be the same as that used n the determnaton of the reserve dstrbuton. Let H be ths arbtrary reserve value. For prcng t would be useful to have C, the expected reserve cost above H (gven that the actual loss s above H). The formula for C s Z 1 C= (x H)f R (x)dx H where R s the random true reserve. Let P be the amount pad and, as before, U s the random, true, ultmate loss. Then R = U Pandso C= = = Z 1 H Z 1 H Z 1 P+H (x H)f R (x)dx (x H)f U (x +P)dx (y P H)f U (y)dy = E(U) E(U ^ P+H) = exp[¹ +ln(v)+¾ 2 =2] " Ã!# ln(h + P) ¹ ln(v) ¾ 2 1 ¾ μ ln(h + P) ¹ ln(v) (H + P) 1 : ¾ The second lne s a change of varable usng R=U P. The same concept apples n stuatons where one company acqures or merges wth another company or a rensurer acqures the reserve of the cedng carrer (loss portfolo transfer). Another way to determne the charge s to calculate the stress reserve value for a gven percentle. Ths can be done easly snce we know the reserve dstrbuton. The dfference between ths and the held reserve s the reservng captal or the commutaton charge. In these cases, C or the full captal s lkely conservatve and a certan fracton may be more approprate as a charge Insurance market segmentaton The expected excess cost formula gves an nsght nto rensurance economcs. The assumng carrer wll pool the assumed reserves nto exstng homogenous reserve segments. If the poolng decreases the total rsk, captured n the estmated ¾ 2 for the lne, the assumng carrer s expected excess cost wll be lower than cedng carrer s. Ths assumes that the reserve adequacy of the reserve segment s estmated dentcally by both partes. The above argument explans nsurance market segmentaton. Companes grow ther busness n a gven lne and contnue to acqure busness from smaller companes n the same lne because they face dfferent total rsks. Unlke other busnesses, the volatlty of losses underlyng estmates drves decsons to acqure and grow an exstng busness segment. Ths s also the reason why low-rsk (short-taled, fast-developng) lnes are seldom ceded to other companes Net and gross reserve dstrbutons It s possble to quantfy the reserve dstrbuton net of rensurance and/or recoveres usng the method explaned n the paper because we arefollowngthenetreserverevewsandsmply measurng the uncertanty n the estmates. One caveat when dealng wth net dstrbutons s that the true mean can be harder to estmate, especally f treates have changed recently. As stated earler, quarterly error trangles are more helpful n such cases because they are more responsve to changes. In other cases, the current actuary s estmate can be taken to be unbased untl further hstory s developed under the new treaty. If rensurance has a relatvely small mpact on total reserves, a change n treaty provsons wll not change the resultng net error dstrbuton sgnfcantly. For example, quota share rensurance on a loss occurrng bass on a lne wth large 44 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 1

16 Quantfyng Uncertanty n Reserve Estmates clam count per accdent year wll not necessarly lead to a lower varablty of the net aggregate error dstrbuton. The reservng actuary wll see proportonally lower losses for new accdent years but ths may not mpact the net error dstrbuton. Another example s casualty excess of loss coverage that attaches at a hgh layer. If the ground up clam count s large enough, the change of rensurance treaty many have very lttle mpact on the net error dstrbuton as few losses out of the total wll be ceded n that layer. In some cases, especally for low-frequency and hgh-severty lnes such as personal umbrella excess of loss coverage, the mpact of a change n treaty can be sgnfcant and the error hstory wll not be relevant to the current treaty. Ths wll requre the actuary to conduct net hstorcal reserve revews based on the new treaty. Ths can be tme consumng but should be done f the lne forms a sgnfcant part of the total reserve. Net dstrbutons result n net reserve ranges and net reservng captal. These are relevant from a solvency, regulatory, and company ratng standpont Regulatory and ratng agency applcatons Regulators and ratng agences are nterested n quantfyng reserve ranges, percentles, and reservng captal n order to montor the solvency of the company. The regulator n partcular s nterested n measurng the performance of the held reserve, shown n Schedule P. Once the dstrbuton of the reserves s known, we can state the percentle of the held reserves. Note that the management bas wll now make a dfference, as t wll poston the held reserves to the rght or left of the mean. We can also calculate the reservng rsk captal as the dfference between a selected cutoff value (say 95th percentle) of the ndcated reserves and the held reserves. The resultng reservng rsk captal can be used to modfy the RBC reservng rsk charge. Note that the NAIC tests of reserve development to surplus ratos suggest comparng dollar ultmate loss errors to company surplus. Ths s very smlar to our approach of measurng reserve uncertanty. Thus, the reservng rsk charge measured usng the method outlned n ths paper would be consstent wth the current annual statement and NAIC practces. 11. Concluson Ths paper presents a shft n paradgm from loss dstrbutons of the underlyng losses to dstrbutons of the company s estmates. Ths represents a new way of measurng reservng rsk. By beng able to quantfy rsk both by lne and for the company, effectve management of captal, rensurance, and other company functons becomes feasble. We provde a framework for assessng reserve revew accuracy as well as measurng the dstrbuton of the current reserve revew. Ths s done usng a dstrbuton revew mmedately afterareserverevewusngthesamesegments and models currently used by the reservng department. Gven the current complex reservng envronment where reservng s both an art and a scence, there s no statstcal formula to set reserves or ts dstrbuton. Rather we present a rgorous framework that nvolves the same consderatons and process as the underlyng reserve revew tself. Acknowledgments We are ndebted to Zhongzan Han (Statstcs and Actuaral Scence Department of the Unversty of Central Florda) for hs assstance wth the lterature revew. The authors would also lke to thank Wllam J. Gerhardt (Natonwde Insurance) for beng a great soundng board n the formatve stages of the paper. VOLUME 4/ISSUE 1 CASUALTY ACTUARIAL SOCIETY 45

17 Varance Advancng the Scence of Rsk The authors are also grateful to an anonymous revewer for extensve data testng and feedback. References Bornhuetter, R. L., and R. Ferguson, The Actuary and IBNR, Proceedngs of the Casualty Actuaral Socety 59, 1972, pp Brown, R. L., and L. R. Gottleb, Introducton to Ratemakng and Loss Reservng for Property and Casualty Insurance (2nd ed.), Wnsted, CT: ACTE, CAS Workng Party on Quantfyng Varablty n Reserve Estmates, The Analyss and Estmaton of Loss and ALAE Varablty: A Summary Report, Casualty Actuaral Socety Forum, Fall 2005, pp De Alba, E., Bayesan Estmaton of Outstandng Clams Reserve, North Amercan Actuaral Journal 6, 2002, pp England, P. D., and R. J. Verrall, Stochastc Clams Reservng n General Insurance, Brtsh Actuaral Journal 8, 2002, pp Fnger, R. J., Modelng Loss Reserve Developments, Proceedngs of the Casualty Actuaral Socety 63, 1976, pp Hayne, R. M., An Estmate of Statstcal Varaton n Development Factor Methods, Proceedngs of the Casualty Actuaral Socety 72, 1985, pp Hayne, R. M., Estmatng and Incorporatng Correlaton n Reserve Varablty, Casualty Actuaral Socety Forum, Fall 2004, pp Mack, T., Dstrbuton-Free Calculaton of the Standard Error of Chan Ladder Reserve Estmates, ASTIN Bulletn 23, 1993, pp Shapland, M., Loss Reserve Estmates: A Statstcal Approach for Determnng Reasonableness, Varance 1, 2007, pp Taylor, G. C., Loss Reservng: An Actuaral Perspectve.Dordrecht, Netherlands: Kluwer Academc Publshers, Taylor, G. C., and F. R. Ashe, Second Moments of Estmates of Outstandng Clams, Journal of Econometrcs 23, 1983, pp Taylor, G. C., and G. McGure, Loss Reservng wth GLMs: A Case Study, Casualty Actuaral Socety Dscusson Paper Program, Applyng and Evaluatng Generalzed Lnear Model, 2004, pp Verrall, R. J., On the Estmaton of Reserves from Loglnear Models, Insurance: Mathematcs and Economcs 10, 1991, pp Verrall, R. J., A Bayesan Generalzed Lnear Model for the Bornhuetter-Ferguson Method of Clams Reservng, North Amercan Actuaral Journal 8:3, 2004, pp CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 1