Parameter Uncertainty in Loss Ratio Distributions and its Implications

Transcription

1 ad its Implicatios Michael G. Wacek, FCAS, MAAA Abstract This paper addresses the issue of parameter ucertaity i loss ratio distributios ad its implicatios for primary ad reisurace ratemakig, uderwritig dowside risk assessmet ad aalysis of slidig scale commissio arragemets. It is i some respects a prequel to Va Kampe s 003 CAS Forum paper [1], which described a Mote Carlo method for quatifyig the effect of parameter ucertaity o expected loss ratios. He showed the effect was especially sigificat i pricig applicatios ivolvig the right tail of the loss ratio distributio. While Va Kampe focused purely o the objective of quatificatio, this paper develops the fuctioal form of the loss ratio distributio icorporatig parameter ucertaity that is implicit i his approach. This paper thus both uderpis Va Kampe s work ad allows us to apply it more efficietly, because it is easier to work with the loss ratio distributio directly tha to perform Va Kampe s simulatio. Suppose we have a set of o-level loss ratios from a stable portfolio of busiess of substatial eough size that it is plausible that the loss ratios ca be viewed as a sample arisig from a approximately ormal or logormal distributio, the parameters of which are ukow. What is the distributio of the prospective loss ratio? This paper discusses the drawbacks of usig the best fit ormal or logormal distributio to model the loss ratio, particularly for pricig or risk assessmet applicatios that deped o the tails of the distributio. While oe fit is best, frequetly a umber of parameter sets provide early as good a fit. Choosig oly the best fit distributio meas igorig the iformatio cotaied i the sample about the other possible distributios. That iformatio ca be reflected i the loss ratio distributio by weightig together all the plausible ormal or logormal distributios, give the sample, by their relative likelihoods. I the cotiuous case, where the weightig fuctio is the desity fuctio of the parameters, the resultig distributio is the Studet s t or log t distributio, respectively. This distributio, which icorporates the ucertaity about the parameters, is preferable to the best fit distributio for modelig the prospective loss ratio. The paper illustrates applicatios ragig from aggregate excess reisurace pricig to measuremet of uderwritig dowside risk to estimatio of the expected cost or beefit of slidig scale commissios, i each case comparig the results arisig from uderlyig ormal ad logormal assumptios ad both parameter certaity ad parameter ucertaity. Keywords: Parameter ucertaity, aggregate loss, aggregate excess, logormal, Studet s t, dowside risk 1. INTRODUCTION This paper addresses the issue of parameter ucertaity 1 i loss ratio distributios ad its implicatios for actuarial applicatios. Very few CAS papers have dealt with the subject of parameter ucertaity, otably Va Kampe [1], Meyers [], [6], Kreps [3], Haye [4] ad Major [5]. The umber is small compared to the dozes of papers that have discussed 1 Sometimes also referred to as parameter risk 165

2 methods of addressig process risk. I fact, there may be more papers cotaiig caveats sayig they do ot deal with parameter risk tha there are papers that address it! I the view of this author the subject deserves more attetio. As actuaries develop icreasigly sophisticated models of risk processes, it is critical that we take accout of our lack of kowledge of the true parameters of these models. Failure to do so ca lead to systematic overcofidece ad wrog coclusios. This paper was ispired by Va Kampe s 003 CAS Forum paper, Estimatig the Parameter Risk of a Loss Ratio Distributio, [1] i which he preseted a Mote Carlo simulatio based approach for quatifyig the impact of parameter risk i certai applicatios. Both his presetatio of the problem ad his solutio were refreshigly clear. Ufortuately, i practice his simulatio approach is a cumbersome oe. This paper develops the fuctioal form of the loss ratio distributio icorporatig parameter ucertaity that is implicit i Va Kampe s approach. It thus both uderpis his work ad allows us to apply it more efficietly, because is it easier to work with the loss ratio distributio directly tha to perform the simulatios. 1.1 Orgaizatio of Paper The paper is orgaized ito six sectios. The first sectio is the Itroductio, where we describe the geeral framework. I the cotext of a give set of loss ratio experiece that has bee adjusted to the prospective claim cost ad rate levels, we defie the prospective loss ratio desity f x (x) as the itegral of the product of the coditioal loss ratio desity f x (x θ) ad the joit desity fuctio of the parameters f θ (θ). Sectio itroduces the assumptio that the coditioal loss ratio distributio is ormal, which allows us to use results from ormal samplig theory to describe the desities of the parameters. We discuss the drawbacks of choosig the best fit ormal distributio f F x (x) as the model of the loss ratio distributio i light of the ucertaity i the best fit parameters, especially i the case of small sample sizes. I Sectio 3 we show how to icorporate parameter ucertaity by applyig the geeral framework described i Sectio 1 to the ormal sceario itroduced i Sectio. We show that the result is a Studet s t desity. We also show how that Studet s t desity ca be approximated as a weighted average of ormal desities, where the weights are discrete 166 Casualty Actuarial Society Forum, Fall 005

3 probabilities associated with the parameters of the plausible ormal desities, which we ca estimate from the iformatio cotaied i the loss ratio experiece. I Sectio 4 we chage the assumptio about the form of the coditioal desity to logormal. Because the logormal desity ca be derived from the ormal by a simple chage of variable, we ca easily determie the formulas for icorporatio of parameter ucertaity i the logormal case from the formulas developed i Sectio 3. The resultig distributio is a log t, which is the Studet s t aalogue to the logormal. We compare the best fit logormal ad the log t. I Sectio 5 we illustrate the four models (ormal ad logormal uder coditios of parameter ucertaity ad parameter certaity ) i the cotext of three applicatios: 1) aggregate excess pricig, ) dowside risk measures, ad 3) slidig scale commissios. Sectio 6 cotais the Summary ad Coclusios, where we recap the mai objectives of the paper, which are described as: 1) demostratig how to derive ad use the desity fuctio of the prospective loss ratio f x (x) i pricig ad risk assessmet applicatios, give o-level loss ratio experiece ad a ormal or logormal loss ratio process, ad ) showig, maily by meas of examples, that f x (x) has fatter tails tha the best fit alterative f F x (x), which implies greater loss exposure i high aggregate excess layers ad greater exposure to frequecy ad severity of uderwritig loss tha that idicated by f F x (x). 1. Framig the Problem Suppose we have accidet years of loss ratio experiece from a stable portfolio of busiess, where the loss ratios have bee adjusted to the projected future claim cost ad rate levels. Assumig the o level adjustmets have bee made perfectly ad the accidet years are idepedet, we ca treat the loss ratio observatios as a radom sample arisig from the stochastic process goverig the geeratio of loss ratios from this portfolio. Let x represet the radom variable for the prospective loss ratio ad let x 1, x, x 3,..., x deote the observed loss ratios. The the sample mea is x = variace is s (x = i x ). i=1 1 x i i= 1 ad the ubiased sample I the basic actuarial ratemakig applicatio, we eed to determie the mea of the prospective loss ratio distributio E ( x ). If x is symmetrically distributed about the mea, Casualty Actuarial Society Forum, Fall

4 the we kow E ( x ) = x. If all we eed is E ( x ), the we do t eed to kow ay more about x. O the other had, if x is ot symmetrically distributed about the mea, the ot oly is E( x ) x, but to determie its value it is ecessary to evaluate x f x (x)dx, which requires kowledge of f x (x). Likewise, i more advaced ratemakig applicatios, e.g., pricig aggregate excess coverage or structurig a loss-sesitive ratig pla, ad i cases where x is ot symmetrically distributed, we eed to kow the distributio of x. I this paper we will discuss how to use o-level loss ratio experiece to determie the distributio of x, give varyig degrees of certaity about the parameters of the uderlyig stochastic process, for the cases where that process is (a) ormal, ad (b) logormal. Because parameter ucertaity ca have a sigificat impact o the ature of the loss ratio distributio, it is critical to the soudess of the pricig (ad reservig) process that such ucertaity is take ito accout. Let θ refer to the set of parameters of the stochastic process that gives rise to the prospective loss ratio. If f x (x θ) is the desity fuctio of the loss ratio, give the parameter set θ, the the margial desity fuctio of x is: fx ( x ) = fx( x θ ) fθ ( θ ) dθ (1.1) θ Formula (1.1) shows that f x (x) ca be see as a weighted average of a set of distributios of the form f x (x θ) where f θ (θ) is the weightig fuctio. If there is o ucertaity about the value of the parameter set, f θ (θ) collapses to a discrete probability fuctio with Pr ( θ ) = 1 ob for θ = θ 0 ad 0 for all other values of θ. I that case f x (x) = f x (x θ 0 ) ad for otatioal coveiece the θ 0 is usually omitted. However, i cases where the values of the parameters are ucertai, care must be take to maitai the distictio betwee f x (x) ad f x (x θ).. x θ NORMALLY DISTRIBUTED Assume x θ is ormally distributed with parameters θ = {µ,σ }, these parameters represetig the populatio mea ad variace, respectively. The values of the parameters The parameter ucertaity regardig the correct distributio family is beyod the scope of this paper. 168 Casualty Actuarial Society Forum, Fall 005

5 are ukow. Treatig these ukow parameters i Bayesia fashio as radom variables, i this cotext formula (1.1) ca be rewritte as: f ( x ) = f ( x µ, σ ) f ( µ, σ )dσ dµ µ σ x x = ( x µ, σ ) f ( µ σ ) f ( σ )dσ d µ σ σ f x µ µ (.1) where f x (x µ,σ 1 ) = σ 1 x µ π e σ (.) is a ormal desity that depeds o µ ad σ. Because x is the ubiased ad maximum likelihood estimator of µ ad s is the ubiased estimator of σ, it is temptig simply to treat µ ad s as parameter costats istead of as radom variables 3, ad set µ = x ad σ = s i formula (.), deem = Pr ob( µ = x ) ad Prob( σ s ) to be close to 1, ad coclude that, for practical purposes, the desity f x (x) ca be approximated by the ormal desity: f F 1 x (x) = s π e 1 x x s (.3) Figure A is a graph of f F x (x) with x = % ad s = The reader might fid it cofusig that we sometimes treat µ ad s as parameter costats ad sometimes as parameter radom variables. However, to avoid overly cumbersome otatio ad discussio that would detract from the coceptual developmet, we will assume the reader ca discer from cotext which form we are discussig. Casualty Actuarial Society Forum, Fall

6 FIGURE A Desity Fuctio f F x (x) Give x = 67.79% ad s = Figure A ad f x F ( x ) represet what is frequetly called the best fit distributio give the sample data. However, we should be cautious about adoptig this distributio as f x (x) without first examiig the error structure of the sample-based parameters, which we will ow do. Give a radom sample of loss ratio observatios, a Bayesia iterpretatio of results from ormal samplig theory allows us to specify the desities f σ (σ ), f µ (µ σ ) ad f µ (µ). 4 We will use those results to examie the risk i the sample-based parameters, begiig with f σ (σ ): f σ 1 ( σ ) = 1 σ Γ( 1 ) ) 1 1 ( 1) s ( 1) s σ e (.4) σ 4 Strictly speakig, we should refer to f σ (σ { x i }), f µ ( µ (σ,{ x i }) ad f µ ( µ { x i }). However, because that otatio is cumbersome ad the coditioality should be clear from cotext, we will drop the referece to the sample { x i }. 170 Casualty Actuarial Society Forum, Fall 005

7 ( 1) Because y 1 = s is a chi-square square radom variable with -1 degrees of σ freedom, the desity represeted by (.4) is sometimes called the iverse chi-square 5. Figure B shows f σ (σ ) graphically for values of equal to 5, 10, 5, ad 100, respectively, give s = The graph for =5 is the most skewed. As icreases, both skewess ad dispersio decreases. The graph for =100 appears early symmetrical. FIGURE B Desity Fuctio f σ (σ ) Give s = , = 5, 10, 5, 100 The mea of σ is a fuctio of whose value approaches s as approaches ifiity: E(σ ) = s 1 3 (.5) A measure of the cofidece we should feel about ascribig to σ a value of s is the probability that σ falls withi a certai tolerace of s. Because we wat to be highly Casualty Actuarial Society Forum, Fall

8 cofidet that σ = s, let s set the tolerace at ±1% of s. Because σ = ( 1) y 1 s, the bouds of this iterval are ( 1) y 1 = (.99) ad ( 1) y 1 = (1.01) ad thus associated with chisquare values, y 1, of ( 1) ( 1) ad (.99) (1.01), respectively. The probability associated with this iterval is F 1 ( 1.99 ) F 1 ( ), where F 1 deotes the chi square cdf with -1 degrees of freedom. The results are tabulated i Table 1, which shows that Prob(. 99 s σ s ) = Prob( σ ) is oly % for =5, risig to 11% for =100. There is very little basis for havig much cofidece i σ = s = ad o basis for claimig total cofidece! TABLE 1 Probability of σ withi +/- 1% of s = 7.71% Give Sample Size Degrees of Freedom Probability σ < 7.63% Probability σ < 7.79% Probability 7.63% < σ < 7.79% % 41.68%.17% % 45.38% 3.3% % 48.89% 5.49% % 53.67% 11.17% Let s ow tur to the distributio of µ. From samplig theory we kow that the desity of µ σ, give a sample of size, is: f µ σ (µ σ ) = 1 σ / π e 1 µ x σ / (.6) 5 See Appedix A for derivatio from the chi square with a chage of variable. 17 Casualty Actuarial Society Forum, Fall 005

9 which is recogizable as a ormal desity. The margial distributio f µ ( µ ) is give by: f µ (µ) = Γ( ) s / ( 1)π Γ( 1 ) µ x s/ (.7) which is a Studet s t desity with -1 degrees of freedom. The mea ad variace of µ are give below as formulas (.8) ad (.9): µ (.8) E ( ) = x Var(µ) = s 1 3 (.9) Figure C shows f µ (µ) graphically for values of equal to 5, 10, 5, ad 100, give x % = ad s = All the graphs are symmetrical about x. The graph for =5 shows the greatest variace ad that of =100 the least. Casualty Actuarial Society Forum, Fall

10 FIGURE C Desity Fuctio f µ (µ) Give x = 67.79%, s = , = 5, 10, 5, 100 By the same reasoig we described for σ, a measure of the cofidece we should feel about ascribig to µ a value of x is the probability that µ falls withi a certai tolerace of x. Because we wat to be highly cofidet that µ = x, let s set the tolerace at ±1% of x. Because, the bouds of this iterval are x t L + s /. 99x ad x + t 1 t U 1 t 1 = µ x s / 1 = U s / = 1.01x. If x = % ad s = , this implies L = t 1 =.01x s / = The cumulative probabilities associated with the upper ad lower bouds are give by T 1 ( ) ad T 1( ) = 1 T 1( ), respectively, where T 1 is the Studet s t cdf with -1 degrees of freedom, which meas that Prob(. 99x µ 1. 01x ) = T 1( ) 1. The results are tabulated i Table, which shows that Prob(. 99x µ 1. 01x ) = Prob( µ. 6847) is 15% for =5, risig to 6% for =100. While this is better tha the case for σ, it still suggests that placig total cofidece i µ = x = % is uwise, particularly for small values of. 174 Casualty Actuarial Society Forum, Fall 005

11 It should be clear from Figures B ad C that the best fit parameters are far from the oly reasoable choice, give the loss ratio experiece. Why ot icorporate iformatio about those other reasoable parameter choices i our determiatio of f x (x)? TABLE Probability of µ withi +/- 1% of x = 67.79% Give Sample Size Degrees of Freedom Probability µ < 67.11% Probability µ < 68.47% Probability 67.11% <µ < 68.47% % 57.31% 14.63% % 60.64% 1.7% % 66.79% 33.59% % 80.93% 61.86% 3. INCORPORATING PARAMETER UNCERTAINTY NORMAL CASE 3.1 Exact Desity I the previous sectio we showed that, especially i small sample cases, it is wrog to treat the fitted distributio f F x (x) give by (.3) as the distributio of x, because there is too great a probability of sigificat variatio i the true value of the parameters from the best fit parameters. There are too may other good parameter choices to be sure that a sigle set of parameters adequately captures all the importat iformatio from that sample. I this sectio, we show how to use the results from samplig theory outlied i the previous sectio together with the iformatio i the sample to obtai the correct characterizatio of f x (x). Casualty Actuarial Society Forum, Fall

12 We ca express the radom variables x µ,σ ad µ σ i formulas (.) ad (.6) i terms of the stadard ormal radom variable z as follows 6 : x µ,σ = µ + z 1 σ (3.1) µ σ = x + z σ / (3.) The radom variable σ described i (.4) ca be expressed as: σ = ( 1) y 1 s (3.3) where y 1 is chi-square with -1 degrees of freedom. Expadig formula (3.1) by replacig the parameter µ with the radom variable µ σ give i formula (3.), we see that: x σ = ( x + z σ / )+ z 1 σ (Because µ σ = x + z σ / ) = x + (z 1 + z / ) σ = x + z σ +1 (Because (z 1 + z / ) = z +1 ) (3.4) Formula (3.4) implies the ormal desity f x (x σ ) give below as formula (3.5), which depeds o σ but ot o µ: 6 Subscripts are used to distiguish the separate istaces of z i formulas (3.1) ad (3.). 176 Casualty Actuarial Society Forum, Fall 005

13 f x (x σ ) = 1 σ +1 e π 1 x x σ +1 (3.5) We ca alteratively expad (3.1) by replacig the parameter σ with the radom variable σ give i formula (3.3) to obtai: x µ = µ + z 1 s (Because z 1 σ = z 1 s ) y y 1 1 = µ + t 1 s (Because z 1 y = t 1 ) (3.6) 1 where t 1 is the stadard Studet s t with -1 degrees of freedom. Formula (3.6) implies the Studet s t desity f x (x µ ) that depeds o µ but ot o σ, give below as formula (3.7): Γ( f x (x µ ) = ) s ( 1)π Γ( 1 ) x µ s (3.7) Returig to (3.4), if we ow expad that formula by replacig the parameter σ with the radom variable σ described i (3.3), we see that: x = x + z s y 1 +1 (Because z σ = z s ) y 1 = x + t 1 s +1 (Because z y 1 = t 1 ) (3.8) Formula (3.8) implies the Studet s t desity f x (x) that depeds o either µ or σ : Casualty Actuarial Society Forum, Fall

14 f x (x) = s Γ( ) +1 1 ( 1)π Γ( 1+ 1 ) 1 s x x +1 (3.9) This is a Studet s t with -1 degrees of freedom, mea of x ad variace of: Var(x) = s (3.10) Figure D shows f x (x) graphically for values of equal to 5, 10, 5, ad 100, respectively, give x = % ad s = All the graphs are symmetrical about x. The graph for =5 shows the greatest variace ad that of =100 the least, with =10 ad =5 i betwee. The graph correspodig to =100 is visually idistiguishable from the graph of a ormal desity with mea 67.79% ad variace (though the former has a slightly larger variace of ). FIGURE D Desity Fuctio f x (x) Give x = 67.79%, s = , = 5, 10, 5, Casualty Actuarial Society Forum, Fall 005

15 3. Approximate Desity Note that formula (3.9) is the result of simplifyig formula (.1) by itegratig over µ ad σ. We ca achieve a approximatio to that itegratio by replacig the desities f µ (µ σ ) ad f σ (σ ) i (.1) with discrete probability weights i the followig summatio: f x (x) f * x (x) = f x (x µ ij,σ j ) p( µ i σ j ) p(σ j ) i j 1 = π e σ j p( µ i σ j ) p(σ j ) (3.11) i j σ j 1 x µ ij where p(µ i σ j ) = p(σ j ) = p(µ i σ j ) p(σ j ) = 1 i j i j Assumig the aalyst has access to software to do umerical or exact itegratio, for most applicatios it is both easier ad more accurate to work directly with f x (x) as defied by formula (3.9) rather tha with the approximatio f * x (x) give by formula (3.11) 7. However, we believe it is istructive to use formula (3.11) to illustrate how the Studet s t desity defied by (3.9) ca be costructed as a weighted sum of ormal desities. We will illustrate the case of =5 with sample mea ad variace of x = % ad s = First, let us divide the domais of each of f ( σ ) ad f σ µ (µ σ ) ito 5 itervals associated with the followig quatiles: 0, 0.04, , , 0.96 ad 1. This results i itervals of legth 0.04, , , ad 0.04, which we will use as weights for the values of σ ad µ σ associated with each iterval. The midpoits of these itervals are 0.0, , 0.50, ad We associate a value of σ with each iterval such that F σ (σ j ) = midpt( j), which implies: σ j = F 1 σ (midpt( j)) ( 1) = 1 (midpt( j)) s (3.1) Y 1 7 We have used CalculatioCeter by Wolfram Research to perform the itegral calculatios for this paper. Casualty Actuarial Society Forum, Fall

16 1 where Y 1 (midpt( j)) represets the chi-square iverse distributio fuctio (with -1 degrees of freedom) evaluated at the midpoit of the j-th iterval. Similarly, we associate a value of µ σ with each iterval such that F µ σ (µ i ) = midpt(i), which implies: µ i σ 1 j = F µ σ (midpt(i)) = x N 1 (midpt(i)) σ j / (3.13) where N 1 (midpt(i)) represets the stadard ormal iverse distributio fuctio evaluated at the midpoit of the i-th iterval. Because µ is depedet o σ, there are five values of µ σ for each µ-related iterval i, oe for each of the values of σ. The results are summarized i Table 3, which show the parameters for 5 ormal distributios ad their associated probability weights. The iterval midpoits F σ (σ j ) ad the correspodig σ j are show i the first two colums. 8 The iterval midpoits F µ σ (µ i ) are displayed across the top of the table with the correspodig µ i σ j show i the body of the table below them. The probability weights associated with each row ad colum are at the right ad bottom of the table respectively. Each value of σ j i the secod colum is to be paired with each of the values of µ i σ j to its right. These parameter pairs defie the ormal distributios to be weighted usig formula (3.11). For example, σ 1 = 4.51% is paired with each of 63.64%, 66.04%, 67.79%, 69.54% ad 71.94% to form (µ,σ ) parameter pairs (4.51%, 63.64%), (4.51%, 66.04%), (4.51%, 67.79%), (4.51%, 69.54%) ad (4.51%, 71.94%), with associated weights of 4% 4%, 4% 30.67%, 4% 30.67%, 4% 30.67% ad 4% 4%, respectively. 8 We display σ j rather tha σ j for presetatioal reasos. 180 Casualty Actuarial Society Forum, Fall 005

17 TABLE 3 Parameters ad Weights for Normal Desities i f * x ( x ) Approximatio Example with =5, x =67.79%, s = Iterval Midpoits F( µ σ ) Iterval Row Midpt F( σ ) σ µ σ Weights % 63.64% 66.04% 67.79% 69.54% 71.94% 4.00% % 6.05% 65.37% 67.79% 70.1% 73.53% 30.67% % 60.06% 64.53% 67.79% 71.05% 75.5% 30.67% % 56.63% 63.09% 67.79% 7.49% 78.95% 30.67% % 46.18% 58.68% 67.79% 76.90% 89.40% 4.00% Colum Weights 4.00% 30.67% 30.67% 30.67% 4.00% Figure E shows this composite desity f * x (x) based o (3.11) ad represeted i Table 3 to be visually idetical to the Studet s t desity f x (x) defied by 3.9 for =5. FIGURE E Desity Fuctios f x (x) ad f x * (x) Give x =67.79%, s = , = 5 Casualty Actuarial Society Forum, Fall

18 A visual fit is ot, of course, adequate for aalytical purposes. Accordigly, if the composite desity is goig to be used for aalysis, the umber ad legth of the itervals should be chose i such a way that the mea ad variace of f * x (x) ad f x (x) match. Matchig meas is a trivial process. Matchig variaces is more complicated. Fortuately, there is a relatioship betwee Var(x), Var(µ) ad E(σ ) that we ca use to facilitate this process: Var(x) = s = s (1 + 1 ) 1 3 = s s = E ( σ ) + Var( µ ) (3.14) This meas we ca test the match betwee Var(x) ad Var(x) * by separately comparig Var(µ) with Var(µ) * ad E(σ ) with E(σ ) * (the asterisks deotig the values of the fuctios based o the discrete approximatio). For =5, exact calculatios give Var(µ) = = ad E( σ ) 5 = = , yieldig a total Var(x) of (or ). This compares to Var( µ ) * = , E(σ ) * = ad Var( x ) * = (or ) based o the approximatio defied i Table 3. Because Var(x) * is oly about 83% of Var(x), this suggests the approximatio could (ad should) be improved by icreasig the umber of itervals ito which the domais of each of µ σ ad σ are divided. However, because our itet was oly to illustrate a simple implemetatio of the approximatio formula (3.11), we will ot pursue the optimizatio of that approximatio here. 3.3 Sectio Summary We ca summarize about how varyig degrees of kowledge about the parameters are reflected i the applicable probability distributio as follows: If both µ ad σ are kow, the f x (x µ,σ ) is a ormal desity with z = x µ σ. 18 Casualty Actuarial Society Forum, Fall 005

19 If oly the value of σ is kow, the f x (x σ ) is a ormal desity with z = x x If oly µ is kow, the f x (x µ ) is a Studet s t desity with t 1 = x µ. s If either µ or σ are kow, f x (x) is a Studet s t desity with t 1 = x x s +1 σ +1 Table 4 shows the 90 th percetile loss ratios correspodig to these kowledge scearios, give x = 67.79% ad s = ad sample sizes ragig from 5 to 100. Several observatios ca be made. First, from row 1 we see that sample size does ot matter if we have certaity about both µ ad σ. Secod, because the loss ratios i row are always less tha those i row 3, it appears that if oly oe of µ or σ ca be kow, it is more helpful to kow σ. Third, we ca see that as the sample size grows larger, f F x (x)= f x (x µ = x,σ = s ) becomes a icreasigly better approximatio of f x (x) at the 90 th percetile... TABLE 4 90 th Percetile of Loss Ratio Distributio* Give x = 67.79% ad s = 7.71% = 5 = 10 = 5 = 100 f x (x µ = x,σ = s ) 77.67% 77.67% 77.67% 77.67% f x (x σ = s ) 78.61% 78.15% 77.87% 77.7% f x (x µ = x ) 79.61% 78.45% 77.95% 77.74% f x (x) 80.74% 78.97% 78.15% 77.79% The 90 th percetile of the weighted ormal approximatio f * x (x) illustrated i Table 3 ad Figure F for =5 is 80.30%, which is close to the true f x (x) value of 80.74%. Further Casualty Actuarial Society Forum, Fall

20 accuracy could be achieved by refiig the umber ad weights of the ormal desities used i the approximatio. 4. INCORPORATING PARAMETER UNCERTAINTY WHEN x θ IS LOGNORMALLY DISTRIBUTED Suppose x θ is logormally distributed with ukow parameters θ = { µ,σ } 9. The the desity of x θ is: f x (x µ,σ ) = xσ 1 π e 1 lx µ σ (4.1) The logormal distributio gets its ame from the fact that w θ = l x θ is ormally distributed with mea µ ad variace σ : f w (w µ,σ 1 ) = e σ π 1 w µ σ (4.) Let w 1,w,w 3,...,w deote the atural logarithms of the respective observed loss ratios x 1,x,x 3,..., x. The the sample log mea is w = w i ad the ubiased sample log variace (w is s w = i w ). i=1 1 We ca use formula (3.9) to determie the margial distributio of w: i=1 Γ( f w (w)= ) s w ( 1)π Γ( ) w w +1 s w (3.9) which, with the chage of variable w = l x, ca be restated as a fuctio of x: f x (x)= f w (w) dw dx 9 Note these parameters take o differet values i the logormal case from their values i the ormal case. 184 Casualty Actuarial Society Forum, Fall 005

21 Γ( = ) xs w ( 1)π Γ ( ) lx w +1 s w (4.3) This log t desity bears the same relatioship to the Studet s t as the logormal does to the ormal. I the same way, we ca use formulas (3.5) ad (3.7) together with the chage of variable w = l x to determie the desities f x (x σ ) ad f x (x µ ): f x (x σ ) = 1 xσ +1 e π 1 l x w σ +1 (4.4) f x (x µ ) = xs w Γ( ) ( 1)π Γ( 1 ) l x µ s w (4.5) Formula (4.4) is a logormal desity. Formula (4.5) is a log t desity. If we igore parameter ucertaity, the best fit parameters of µ = w ad σ = s w imply the desity: f x F (x)= xs w 1 π e 1 lx w s w (4.6) which is the logormal aalogue to formula (.3). Casualty Actuarial Society Forum, Fall

22 As we did i the case of the ormally distributed x θ, we agai cousel cautio before adoptig this best fit logormal f F x (x)as the correct characterizatio of f x (x), because it does ot accout for ucertaity i the parameters. FIGURE F Desity Fuctios f x ( x ) ad f F x (x) Give w = , s w = , = 5 Figure F is a graph of the log t desity f x ( x ) defied by formula (4.3) with =5, plotted together with the best fit logormal desity f x F ( x ) defied by (4.6). Values of = ad s w = were determied from the same data sample that yielded w x = % ad s = used i the examples of Sectio 3. The log t distributio clearly has a larger variace ad is slightly more skewed tha the best fit logormal. A aalyst relyig o the best fit logormal to draw coclusios about the behavior of x, especially i the tails, will uderestimate the likelihood of occurreces of x i the tails. The log t desity represetig f x ( x ) ca be approximated as a weighted average of logormal desities by usig formula (3.11) with the ormal desity replaced with the aalogous logormal desity. I practice, it is usually easier to umerically itegrate the log t directly tha to costruct ad the itegrate the equivalet composite desity. 186 Casualty Actuarial Society Forum, Fall 005

23 Oe drawback to formula (4.3) is that E ( x ) ad Var ( x ) are ifiite i realistic scearios where is small ad/or s is ot small. 10 For example, if w = ad s w = , E ( x ) is ifiite i the case of =5. I practice, this is ot as bad as it souds. If 1 Fx ( x f ( x )dx is a plausible mea value of x, we ca coclude that the o-covergece ) 0 x of x f x ( x ) dx is due to behavior i the extreme right tail of f x (x). For practical purposes it F is safe to approximate the mea of x as 1 x (.9999) E( x) = xf x (x)dx. For example, i the =5 0 case just cited, F x (.9999 ) = 346% ad xf x (x)dx= 68.43%, which is a plausible value for the mea. 0 A implicatio of the assumptio that x θ is logormally distributed that we do ot fully uderstad is that the value of E( x) = x fx ( x) dx calculated directly usig the desity fuctio 0 exceeds the sample mea x. We fid it puzzlig because (a) x is the ubiased estimator of the mea of ay distributio ad (b) f x (x) was parameterized usig the ubiased estimators w ad s w for µ ad σ, respectively. It seems both should be correct, ad yet they do ot match. I the example we have bee followig, where x = 67.79%, eve usig the logormal desity give i formula (4.6), which implies o parameter ucertaity, we obtai E( x ) = %. Whe we allow for parameter ucertaity (implyig use of the log t desity give by (4.3)), the uderestimatio of E ( x ) by x icreases. I particular, for = 5, 10, 5 ad 100, respectively, E ( x ) equal to 68.43% 11, 68.0%, 67.90% ad 67.85%, implyig differeces of 0.64, 0.3, 0.11 ad 0.06 loss ratio poits, respectively. The differece is particularly oteworthy for =5. 5. APPLICATIONS 5.1 Experiece Loss Ratios I this sectio we illustrate the applicatio of the foregoig to real world problems, i particular, to the pricig of aggregate excess reisurace, the assessmet of uderwritig 10 We draw that coclusio because our attempt to umerically itegrate x f x ( x ) dx did ot coverge to a solutio. 11 F 1 x (.9999) Calculated as xf x (x)dx, because xf x (x)dx does ot coverge. 0 0 Casualty Actuarial Society Forum, Fall

24 dowside risk ad the determiatio of expected commissios uder slidig scale arragemets. Suppose we have bee give 5 years of o-level loss ratios x i ad their logs w i = l x i, which are show i Table 5 1. Exposure has bee costat over the experiece period. The sample meas, variaces ad stadard deviatios based o equal weightig of the data poits are show at the bottom. We kow that the historical portfolio was large eough that it is plausible that each year s loss ratio arises from a approximately ormal distributio. However, it is also plausible that the loss ratio distributio has some residual skewess, which meas a logormal model might be appropriate. TABLE 5 O-Level Loss Ratio Experiece Accidet Year Weight x i l xi 1 0% 66.95% % 59.68% % 76.41% % 7.5% % 77.79% Mea 70.67% Variace* 0.554% St. Dev.* 7.45% * Ubiased, i.e., E(s ) = σ. For the applicatios illustrated i this sectio we will use four models for f x (x) based o: (1) ormal ad () logormal assumptios for x θ uder coditios of: (A) parameter ucertaity ad (B) parameter certaity. 1 The loss ratios i Table 5 were draw from a logormal distributio with parameters µ = ad σ =.0998, but let us assume we do ot kow that. 188 Casualty Actuarial Society Forum, Fall 005

25 Give the experiece i Table 5, if we assume x θ is ormally distributed, the f x (x) is give by formula (3.9) with x = 70.67% ad s = 7.45%. Alteratively, if we assume x θ is logormal, the f x ( x ) is give by formula (4.3) with w = ad s w = O the other had, if we assume x θ is ormal ad we believe µ = x = % ad σ = s = 7.45% with certaity, the we must use f x (x)= f F x (x) as give by formula (.3). Similarly, if we believe x θ is logormally distributed with µ = w = ad σ = s w = with certaity F we must use fx ( x ) = fx ( x ) as give by formula (4.6). These four model choices ad their characteristics are summarized i Table 6. It is worth poitig out that the logormal-based models A ad B agai both idicate the desitybased value E(x) to be greater tha x. TABLE 6 Summary of Models of ( x ) f x Model f x (x θ) θ f x (x) Formula * E ( x ) A1 Normal Ucertai t % A Logormal Ucertai Log t % B1 Normal Certai Normal % B Logormal Certai Logormal % * Give the loss ratio experiece i Table 5 5. Aggregate Excess Reisurace The pure premium of a aggregate excess layer of L excess of R, where the limit L ad the retetio R are ratios to premiums, is give by: L+R (x R) f x (x)dx+ L f x (x)dx R (5.1) L+R Casualty Actuarial Society Forum, Fall

26 Suppose we are asked to price 0 poits of coverage excess of a 70% loss ratio i four layers of 5% each. Table 7 summarizes the results of usig formula (5.1) with models A1, A, B1 ad B. The models icorporatig parameter ucertaity (A1 ad A) idicate larger pure premiums i every layer tha do the models that assume parameter certaity (B1 ad B). While the differece is modest i the first layer of 5% excess of 70% (o the order of 3% to 4%), it rises rapidly as the retetio icreases. The pure premiums for the fourth layer of 5% excess of 85% for models A1 ad A are respectively 300% ad 00% higher tha from models B1 ad B! Uless the parameters really are kow with certaity, it is foolhardy to use model B1 or B to price aggregate excess layers. TABLE 7 Pure Premiums of Aggregate Excess Layers Give Sample i TABLE 5 Limit 5% 5% 5% 5% Model f x (x θ) θ Retetio 70% 75% 80% 85% A1 Normal Ucertai.09% 1.14% 0.56% 0.8% A Logormal Ucertai.04% 1.17% 0.64% 0.36% B1 Normal Certai.0% 0.9% 0.30% 0.07% B Logormal Certai 1.97% 0.95% 0.37% 0.1% 5.3 Dowside Risk Measures Suppose B represets the isurer s uderwritig breakeve loss ratio. The expected value of the uderwritig result UR is give by: E(UR) = (B x) f x (x)dx (5.) Casualty Actuarial Society Forum, Fall 005

27 E (UR ) ca be expressed as the expected cotributio from uderwritig profit scearios UP > 0 less the expected cost of uderwritig loss scearios UL > 0 : E(UR ) E(UP > 0 ) E(UL > 0 ) = (5.3) E(UP > 0 ) = B (B x) f x (x)dx (5.4) 0 E(UL> 0 ) = (x B) f x (x)dx (5.5) B As the pure premium cost of uderwritig loss scearios, E(UL > 0 ) is a measure of the isurer s uderwritig dowside risk. The probability or frequecy of the isurer icurrig a uderwritig loss UL > 0 is give by: Freq (UL > 0 ) = Pr ob(ul > 0 ) = f x (x)dx (5.6) B The expected severity of uderwritig loss, give UL > 0, is: Sev(UL > 0 ) = E(UL UL > 0 ) = B (x B) f x (x)dx B f x (x)dx = E(UL) Pr ob(ul > 0) (5.7) Note that Sev(UL > 0 ) is the Tail Value at Risk (for uderwritig loss) described by Meyers[] as a coheret measure of risk ad by the CAS Valuatio, Fiace ad Ivestmets Committee[3] for potetial use i risk trasfer testig of fiite reisurace cotracts. Casualty Actuarial Society Forum, Fall

28 We ca use the measures defied by formulas (5.5), (5.6) ad (5.7) to describe the isurer s uderwritig dowside risk. Give a uderwritig breakeve loss ratio of B = 75%, Table 8 shows the results of usig the loss ratio experiece cotaied i Table 5 together with the f x ( x ) models A1, A, B1 ad B discussed i our aalysis of aggregate excess pure premiums. For example, give the assumptio that x θ is ormally distributed with ukow parameters (model A1), there is a probability of 31.19% that the isurer will have a uderwritig loss averagig 7.48 poits. This equates to a expected uderwritig dowside cost of.33 poits. I cotrast, give the assumptio that x θ is ormally distributed with kow parameters based o the loss ratio experiece (model B1), there is a probability of 8.06% that the isurer will icur a uderwritig loss of average severity equal to oly 4.6 poits, which equates to a expected dowside pure premium of 1.30 poits. Similarly, the logormal model icorporatig parameter ucertaity (A) shows much larger measures of frequecy, severity ad dowside pure premium tha the logormal model assumig parameter certaity (B). It should be clear that igorig parameter ucertaity i characterizig dowside uderwritig risk has potetially very serious ad adverse cosequeces for a isurer s uderstadig of the uderwritig risk it has assumed. TABLE 8 Measures of Dowside Risk Give Sample i TABLE 5 Model f x (x θ) θ Freq(UL) Sev(UL) E(UL) A1 Normal Ucertai 31.19% 7.48%.33% A Logormal Ucertai 30.95% 9.6%.87% B1 Normal Certai 8.06% 4.6% 1.30% B Logormal Certai 7.78% 5.34% 1.48% 19 Casualty Actuarial Society Forum, Fall 005

29 5.4 Slidig Scale Commissios Suppose a quota share reisurace treaty has bee egotiated where the cedig commissio is determied accordig to a slidig scale. A miimum commissio of 0% is payable if the loss ratio is 70% or higher. The commissio slides up at a rate of 0.5 poit for every poit of reductio i the loss ratio below 70%, up to a maximum commissio of 5% at a loss ratio of 60% or lower. The expected value of the cedig commissio C ca be expressed by formula (5.8) below: 70% E(C ) = 0% f x (x)dx+ (0% + 70% x 60% f x ( x))dx+ 5% f x (x)dx (5.8) 70% 60% 0 Give the o-level loss ratio experiece i Table 5, what is the expected value of the cedig commissio? We have calculated the expected commissios based o ormal ad logormal assumptios for x θ uder coditios of parameter ucertaity ad certaity (models A1, A, B1 ad B) ad have tabulated the results i Table 9. I all cases the modeled cedig commissios are higher tha the 0% commissio that would be payable at a loss ratio of 70.67%. The differeces rage from 1.0% to 1.4%. The commissios idicated by all the models are clustered very closely together, ragig betwee 1.0% ad 1.4%. Because the cedig commissio slides i respose to loss ratios that are ear E ( x ), where the model differeces are less proouced, the effect of parameter ucertaity is immaterial (at least i this example). Casualty Actuarial Society Forum, Fall

30 TABLE 9 Expected Cedig Commissios Give Sample i TABLE 5 Model f x (x θ) θ 70.67% E(C) Diff A1 Normal Ucertai 0.00% 1.37% 1.37% A Logormal Ucertai 0.00% 1.4% 1.4% B1 Normal Certai 0.00% 1.0% 1.0% B Logormal Certai 0.00% 1.4% 1.4% 5.5 Uequal Loss Ratio Weights The previous examples were based o the assumptio that it is appropriate to weight each observed o-level loss ratio i the historical experiece equally. While that is a coveiet assumptio, it is ot a realistic oe, because exposure teds to chage from year to year. Accordigly, i the iterest of providig additioal examples that are also more realistic, we have tabulated aother set of o-level loss ratios i Table 10. These observed loss ratios arose from the same distributio as the loss ratios i Table 5. The sample mea, variace ad stadard deviatio statistics have bee computed both o a weighted basis ad o the stadard uweighted basis. The formulas for weighted mea ad the ubiased weighted sample variace s c are: c x c = i x i (5.9) c i=1 s c c = i (x i x c ), (5.10) c ( 1) i=1 where c i deotes the weight to be used with the i-th observatio, c is the mea weight ad x c is the weighted mea. 194 Casualty Actuarial Society Forum, Fall 005

31 TABLE 10 O-Level Loss Ratio Experiece d Sample Accidet Year Weight x i l xi 1 16% 53.88% % 53.15% % 70.6% % 73.06% % 56.55% Uweighted Mea 63.45% Variace* 0.744% St. Dev.* 8.6% Weighted Mea 64.00% Variace* 0.767% St. Dev.* 8.76% * Ubiased, i.e., E(s ) = σ. Though the loss ratio experiece show i Table 10 emerged from the same uderlyig loss ratio distributio as that i Table 5, its mea ad stadard deviatio are sigificatly differet. O a uweighted basis the loss ratio mea i Table 10 is more tha 7 poits (more tha 10%) less tha the loss ratio mea i Table 5 (64.00% v %). O the other had, the stadard deviatio is more tha 15% greater (8.6% vs. 7.45%). The sample variatio illustrated by those differeces is worth rememberig whe we are tempted to put great weight o the credibility of a small sample. We have calculated the aggregate excess pure premiums for the layers defied i Table 7 usig the weighted basis loss ratio experiece i Table 10 ad displayed the results i Table 11. As i the example based o Table 5, the pure premiums for all layers are higher whe Casualty Actuarial Society Forum, Fall

32 priced usig the models that icorporate parameter ucertaity (A1 ad A) tha the models assumig the parameters are kow with certaity (B1 ad B). Agai the pricig differece icreases as the retetios icrease. However, it is also worth otig that the differeces i pure premiums based o Table 10 are far less tha the differece betwee those pure premiums ad those calculated based o the experiece i Table 5. For example, i Table 11 we see the idicated model A1 pure premium for 5% excess of 70% is 0.6% compared to.09% i Table 7. The idicated pure premiums for all other layers ad models are also much lower i Table 11 tha i Table 7. Both experiece samples arose from the same loss ratio distributio, but the two samples idicate dramatically differet pure premiums! TABLE 11 Pure Premiums of Aggregate Excess Layers Give Sample i TABLE 10 Limit 5% 5% 5% 5% Model f x (x θ) θ Retetio 70% 75% 80% 85% A1 Normal Ucertai 0.6% 0.46% 0.34% 0.5% A Logormal Ucertai 0.59% 0.46% 0.35% 0.7% B1 Normal Certai 0.51% 0.33% 0.0% 0.1% B Logormal Certai 0.49% 0.33% 0.1% 0.13% Table 1 shows the dowside risk statistics calculated o the basis of the weighted loss ratio experiece i Table 10. Because the sample mea Table 10 is much lower tha i Table 5, the idicated probability of uderwritig loss is much reduced from that show i Table 8. While the severity of uderwritig loss is ot much affected, due to the large reductio i frequecy, the expected cost of uderwritig losses is much lower i Table 1 tha i Table 8. The differece is much greater for the parameter certaity models B1 ad B tha for models A1 ad A. Models B1 ad B ow idicate miimal dowside risk as measured by E (UL ) values of 0.49% ad 0.54%. These compare to values of 1.30% ad 1.48%, respectively, i Table 8, reductios of about two-thirds. O the other had models A1 ad A are less sesitive to the sample variatio. Model A1 s E (UL ) of 1.40% is 40% 196 Casualty Actuarial Society Forum, Fall 005

33 less tha its value i Table 8. The A E (UL ) of 1.87% is about 35% less tha its value i Table 8. Eve at these reduced values both idicate sigificat dowside risk ad both show expected uderwritig loss costs more tha three times as high as B1 ad B. TABLE 1 Measures of Dowside Risk Give Sample i TABLE 10 Model f x (x θ) θ Freq(UL) Sev(UL) E(UL) A1 Normal Ucertai 15.78% 8.86% 1.40% A Logormal Ucertai 15.88% 11.75% 1.87% B1 Normal Certai 11.53% 4.7% 0.49% B Logormal Certai 10.59% 5.06% 0.54% Table 13 shows the expected cedig commissios based o the weighted loss ratio experiece i Table 10. As we saw i the commissios based o the loss experiece show i Table 5 ad displayed i Table 9, there is little variatio i the commissio estimates based o usig the differet models. The expected commissios i Table 9 rage from.65% to.81% compared to a rage of 1.0% to 1.4% i Table 13. The differece due to the variatio i loss ratio experiece is far more importat tha the differece i models. Models A1 ad A show oly about 1.3 poits icrease i expected cedig commissio ad Models B1 ad B show oly about 1.5 poits icrease, eve though the sample loss ratio is more tha 7 poits lower. Casualty Actuarial Society Forum, Fall

34 TABLE 13 Expected Cedig Commissios Give Sample i TABLE 11 Model f x (x θ) θ 64.00% E(C) Diff A1 Normal Ucertai 3.00%.65% (0.35%) A Logormal Ucertai 3.00%.76% (0.4%) B1 Normal Certai 3.00%.7% (0.8%) B Logormal Certai 3.00%.81% (0.19%) 6. SUMMARY AND CONCLUSIONS The mai objectives of this paper have bee to: 1) demostrate how to derive ad use the desity fuctio f x ( x ) of the prospective loss ratio i pricig ad risk assessmet applicatios, give o-level loss ratio experiece ad a ormal or logormal loss ratio process, ad ) show, maily by meas of examples, that f x ( x ) has fatter tails tha the best fit alterative f F ( x ), which implies greater loss exposure i high excess layers ad greater x exposure to frequecy ad severity of uderwritig loss tha that idicated by f F ( x ) x. I distributioal terms, we have show that if we believe the o-level loss ratios are ormally distributed, our lack of kowledge of the parameters of that ormal distributio requires that f x ( x ) be characterized as a Studet s t rather tha a ormal distributio. We may still believe the loss ratio is ormally distributed, but we do ot have sufficiet kowledge to safely characterize it as such. The Studet s t, which does approximate the ormal for large sample sizes (see Figure D), is the best we ca do. Similarly, if we believe the o-level loss ratios are logormally distributed, our lack of kowledge of the parameters of that logormal distributio meas that f x ( x ) must be characterized as a log t rather tha a logormal distributio, for the reasos described above. 198 Casualty Actuarial Society Forum, Fall 005

35 Two other poits also bear repeatig. First, for right-skewed distributios, the sample mea x appears to give a lower estimate of E ( x ) tha the oe determied from the desity fuctio parameterized with ubiased estimators derived from the sample. The differece is less proouced for large sample sizes, but for small experiece samples it is sizeable. We do ot kow what to make of this, but it adds to our discomfort about beig overcofidet about coclusios draw from small samples. Secod, small experiece samples ca exhibit sigificat variatio from the characteristics of the populatio from which they arise, which ca lead to over-pricig or uder-pricig eve whe usig the correct form of ( x ) f x. Actuaries must resist the temptatio to be overcofidet about the ifereces that ca safely be draw from small samples. It is wise to avoid stakig too much o the coclusios of a pricig aalysis based o a small sample. Some further caveats apply. While the methods described i this paper icorporate the cosequeces of our ucertaity about some critical parameters ito estimates of the projected loss ratio, ote that they do ot address other importat sources of parameter ucertaity, ad accordigly, are likely to uderestimate the total variace of x. They address oly the ucertaity arisig from the sample loss ratios, give that those loss ratios are themselves certai. However, those loss ratios are estimates. Therefore, these methods do ot reflect parameter ucertaity associated with loss developmet factors used for the projectio of reported loss ratios to ultimate, or do they reflect ucertaity i the o-level adjustmet parameters. I additio, we do ot kow for certai that we have chose the correct model distributio i the ormal or the logormal. Thus, while this method is a improvemet over methods that do ot icorporate ay parameter ucertaity, a certai amout of cautio remais i order. Casualty Actuarial Society Forum, Fall

36 Appedix A Derivatio of Formula (.4) Assume y 1 is chi square with -1 degrees of freedom. That implies f y (y 1 ) = 1 1 y 1 e 1 y 1 Γ( 1 ) Perform the chage of variable y 1 = ( 1) s, where σ is the ew radom variable. σ The dy dσ = ( 1) (σ ) s ad f σ (σ dy ) = f y (y 1 ) dσ 1 ( 1) = 1 s Γ( 1 ) σ 1 1 e 1 ( 1) s σ ( 1) (σ ) s = 1 σ 1 Γ( 1 ) 1 ( 1) s 1 σ e ( 1) s σ (.4) 00 Casualty Actuarial Society Forum, Fall 005

37 7. REFERENCES [1] Va Kampe, Charles E., Estimatig the Parameter Risk of a Loss Ratio Distributio, Casualty Actuarial Society Forum, Volume: Sprig, 003, p , [] Meyers, Gle G., Parameter Ucertaity i the Collective Risk Model, PCAS LXX, 1983, p , [3] Kreps, Rodey E., Parameter Ucertaity i (Log)Normal Distributios, PCAS LXXXIV, 1997, p , [4] Haye, Roger M., Modelig Parameter Ucertaity i Cash Flow Projectios, Casualty Actuarial Society Forum, Volume: Summer, 1999, p , [5] Major, Joh A., Takig Ucertaity ito Accout: Bias Issues Arisig from Parameter Ucertaity i Risk Models, Casualty Actuarial Society Forum, Volume: Summer, 1999, p , [6] Meyers, Gle G., Estimatig Betwee Lie Correlatios Geerated by Parameter Ucertaity, Casualty Actuarial Society Forum, Volume: Summer, 1999, p. 197-, [7] Meyers, Gle G., The Cost of Fiacig Isurace, Casualty Actuarial Society Forum, Volume: Sprig, 001, p. 1-64, [8] CAS Valuatio, Fiace, ad Ivestmets Committee Accoutig Rule Guidace Statemet of Fiacial Accoutig Stadards No Cosideratios i Risk Trasfer Testig, Casualty Actuarial Society Forum, Volume: Fall, 00, p , Abbreviatios ad otatios CAS, Casualty Actuarial Society C, cedig commissio rate E(UL), expected value cost of uderwritig loss scearios Freq(UL), frequecy of uderwritig loss scearios L, aggregate excess layer limit, i loss ratio poits R, aggregate excess retetio, i loss ratio poits Sev(UL), mea severity of uderwritig loss scearios µ, first parameter of a ormal or logormal distributio, sometimes a radom variable σ, secod parameter of a ormal or logormal distributio, sometimes a radom variable θ, parameter set, umber of years i the loss ratio experiece sample c i, weight for the i-th observed o-level experiece loss ratio c, mea of the weights used with observed o-level experiece loss ratios s, variace of the o-level experiece loss ratios (ubiased) s c, weighted variace of the o-level experiece loss ratios (ubiased) s w, variace of logs of the o-level experiece loss ratios (ubiased) t 1, a Studet s t distributio radom variable with -1 degrees of freedom w, radom var for the log of prospective loss ratio give ucertaity about uderlyig distributio parameters w θ, radom rs variable for the log of prospective loss ratio give parameters of uderlyig distributio w i, log of i-th observatio of o-level experiece loss ratios w, mea of the logs of the o-level experiece loss ratios x, radom variable for the prospective loss ratio give ucertaity about parameters of uderlyig distributio x θ, radom variable for the prospective loss ratio give parameters of uderlyig distributio x i, i-th observatio of the o-level experiece loss ratios x, mea of the o-level experiece loss ratios Casualty Actuarial Society Forum, Fall