CREDIBILITY - PROBLEM SET 1 Limited Fluctuation Credibility

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1 CREDIBILITY PROBLEM SET 1 Limited Fluctuation Credibility 1 The criterion for the number of exposures needed for full credibility is changed from requiring \ to be ithin IÒ\Ó ith probability Þ*, to requiring \ to be ithin 5IÒ\Ó ith probability Þ*& Find the value of 5 that results in no change in the standard for full credibility for number of exposures of \ A) 0524 B) 0548 C) 0572 D) 0596 E) Total claim amount per period W follos a compound Poisson claims distribution The standard for full credibility for total claims in a period W based on number of claims is 1500 claims It is then discovered that an incorrect value of the coefficient of variation for the severity distribution ] as used to determine the full credibility standard The original coefficient of variation used as Þ', but the corrected coefficient of variation for ] is Þ&!! Find the corrected standard for full credibility for W based on number of claims A) 1300 B) 1325 C) 1350 D) 1375 E) The partial credibility factor for random variable \ based on!! exposures of \ is ^ Þ%! Ho many additional exposures are needed to increase the partial credibility factor to at least Þ&!? A) 55 B) 56 C) 57 D) 58 E) 59 4 Total claims per period W follos a compound Poisson distribution and claim severity has the ' pdf 0ÐCÑ&C, for C A full credibility standard based on number of exposures of W needed has been determined so that the total cost of claims per period is ithin 5% of the expected cost ith a probability of 90% If the same number of exposures for full credibility of total cost is applied to the number of exposures needed for the frequency variable R, the actual number of claims per exposure period ould be ithin 100 < % of the expected number of claims per exposure period ith probability 95% Find < A) 054 B) 058 C) 062 D) 066 E) An analysis of credibility premiums is being done for a particular compound Poisson claims distribution, here the criterion is that the total cost of claims is ithin 5% of the expected cost of claims ith a probability of 90% It is found that ith 8'! exposures (periods) and \ )!Þ!, the credibility premium is )*Þ%( After 20 more exposures (for a total of 80) and revised \ )&, the credibility premium is *!Þ)) After 20 more exposures (for a total of 100) the revised \ is )(Þ& Assuming that the manual premium remains unchanged in all cases, and assuming that full credibility has not been reached in any of the cases, find the credibility premium for the 100 exposure case A) 1915 B) 1925 C) 1935 D) 1945 E) 1965

2 6 Total claims per period W has a compound Poisson distribution You have determined that a sample size of 2670 claims is necessary for full credibility for total claims per period if the severity distribution is constant If the severity distribution is lognormal ith mean 1000 and variance 1,500,000, find the number of claims needed for full credibility of total claims per period A) 6650 B) 6675 C) 6700 D) 6725 E) You are given the folloing: The number of claims follos a Poisson distribution The variance of the number of claims is 10 The variance of the claim size distribution is 10 The variance of aggregate claim costs is 500 The number of claims and claim sizes are independent The full credibility standard has been selected so that actual aggregate claim costs per period ill be ithin 5% of expected aggregate claim costs 95% of the time Using the methods of limited fluctuation credibility determine the number of claims required for full credibility of aggregate claim costs per period 8 You are given the folloing: The number of claims per period follos a Poisson distribution Claim sizes follo a lognormal distribution ith parameters (unknon) and 5 % The number of claims and claim sizes are independent 6,600 expected claims are needed for full credibility of aggregate claims per period The full credibility standard has been selected so that actual aggregate claim costs per period ill be ithin 10% of expected aggregate claim costs per period T % of the time Using the methods of limited fluctuation credibility to determine the value of T 9 W has a compound distribution ith frequency R and severity ] R and all claim amounts are independent of one another Limited fluctuation credibility is being applied to W, ith the full credibility standard based on the sample mean of W being ithin 5% of the true mean of W ith probability 90% The folloing information is given regarding the three equivalent full credibility standards for W The expected number of exposures of W needed for full credibility is The expected aggregate amount of claim needed for full credibility is 10,824 The expected total number of claims needed for full credibility is 5412 Find all of the folloing quantities: ß Z +<ÐWÑß IÐRÑ and IÐ] Ñ Þ Sho that R cannot have a Poisson distribution and ] cannot have an exponential distribution

3 10 R is the distribution of the number of claims occuring per eek R has a Poisson distribution ith an unknon mean The standard for full credibility for R is based on the sample mean of R being ithin 5% of the true mean of R ith probability 90% With 400 observed claims in 20 eeks, the credibility premium based on partial credibility is T With 500 observed claims in 30 eeks, the credibility premium based on partial credibility is T Þ* Find the credibility premium based on partial credibility if there are 550 observed claims in 35 eeks Assume that the same manual premium is used in all cases 11 You are given: (i) \:+<>3+6 pure premium calculated from partially credible data (ii) IÒ\ :+<>3+6Ó (iii) Fluctuations are limited to 5 of the mean ith probability T (iv) ^credibility factor Which of the folloing is equal to T? (A) T<Ò 5 Ÿ \:+<>3+6 Ÿ 5 Ó (B) T<Ò^ 5 Ÿ Z\ :+<>3+6 Ÿ ^ 5Ó (C) T<Ò^ Ÿ Z \:+<>3+6 Ÿ^ Ó (D) T<Ò5Ÿ Z\ :+<>3+6 Ð^Ñ Ÿ5Ó (E) T<Ò 5 Ÿ Z\ Ð^Ñ Ÿ 5 Ó :+<> (SOA) You are given: (i) Claim counts follo a Poisson distribution (ii) Claim sizes follo a lognormal distribution ith coefficient of variation 3 (iii) Claim sizes and claim counts are independent (iv) The number of claims in the first year as 1000 (v) The aggregate loss in the first year as 675 million (vi) The manual premium for the first year as 500 million (vii) The exposure in the second year is identical to the exposure in the first year (viii) The full credibility standard is to be ithin 5% of the expected aggregate loss 95% of the time Determine the limited fluctuation credibility net premium (in millions) for the second year (A) Less than 55 (B) At least 55, but less than 57 (C) At least 57, but less than 59 (D) At least 59, but less than 61 (E) At least An insurer has to separate classes of policies The characteristics of the loss per insured in each of the to classes during a one year period are as follos: Class I: Expected claim per insured is 100 To be ithin 5% of expected loss 90% of the time, the standard for number of insureds needed for full credibility is Class II: Expected claim per insured is 200 To be ithin 5% of expected loss 90% of the time, the standard for number of insureds needed for full credibility is Class I has tice the number of insureds as Class II The to classes of insureds are combined and regarded as a single class ith the appropriate adjusted loss per insured during a one year period Find the full credibility standard for the minimum number of insureds required in the combined portfolio, here the full credibility is to be ithin 5% of expected loss 90% of the time (A) (B) (C) (D) (E) 13530

4 14 The partial credibility approach is applied to a data set of 50 claim amounts It is assumed that the claim amount distribution is uniform on the interval Ò!ß ) Ó The full credibility standard is to be ithin 5% of the expected claim amount 90% of the time The partial credibility factor ^ is found After 25 additional claim amounts are recorded, the claim amount distribution is revised to be uniform on the interval Ò!ß Þ) Ó The revised partial credibility factor ^ is found Find the ratio ^Î^ (A) (B) É Þ& Þ& (C) 1 (D) ÈÞ& (E) Þ& 15 Number of claims per year follos a Poisson distribution A number of claims are recorded over a specified period of time A full credibility standard is set so as to be ithin 5% of expected claims per year 90% of the time Based on the observed number of claims, the full credibility standard is not met, but the partial credibility factor is ^Þ') Find the maximum value of 5 so that this same number of claims satisfies a full credibility standard ithin 10% of expected claims per year 5% of the time (A) 995 (B) 99 (C) 98 (D) 975 (E) W has a compound distribution ith frequency R and severity ] R and all claim amounts are independent of one another Limited fluctuation credibility is being applied to W, ith the full credibility standard based on the sample mean of W being ithin 5% of the true mean of W ith probability 90% The folloing information is given regarding the three equivalent full credibility standards for W The expected number of exposures of W needed for full credibility is The expected aggregate amount of claim needed for full credibility is 10,824 The expected total number of claims needed for full credibility is 5412 Find all of the folloing quantities: ß Z +<ÐWÑß IÐRÑ and IÐ] Ñ Þ 17 The aggregate loss in one eek, W, follos a compound negative binomial distribution, and the severity distribution is exponential Limited fluctuation credibility is being applied to W so that the full credulity standard is to be ithin 5% of expected aggregate losses 95% of the time It is found that the expected number of claims needed for full credibility is 5,412 Suppose that the frequency distribution is modified (but still negative binomial) so that mean and variance of the frequency both increase by 20% Find the full credibility standard for the number of claims needed for the ne compound negative binomial distribution (severity is the same exponential distribution as before)

5 18 A compound distribution ÐWÑ has a Poisson frequency distribution ÐR Ñ ith mean For parts (a) and (b), assume that the severity distribution Ð] Ñ is uniform on the interval Ò!ß ) Ó (a) Limited fluctuation credibility is applied to ] based on the sample mean of ] being ithin 5% of the true mean of ] ith probability 90% Find expressions for (i) the expected number of observations of ] needed for full credibility, and (ii) the expected sum of the observed values of ] needed for full credibility (b) Limited fluctuation readability is applied to W based on the sample mean of W being ithin 5% of the true mean of ] ith probability 90% Find expressions for (i) the expected number of observations of W needed for full credibility, and (ii) the expected sum of the observed values of W needed for full credibility, and (iii) the expected total number of claims needed for full credibility 19 [ is a random variable ith mean IÒ[ Ó and variance Z +<Ò[ Ó In a partial credibility analysis of [, the manual premium used is Q!!! A sample of 350 observations of [ is available and the sum of the observed values is 300,000 Partial credibility is applied to determine a credibility premium based on the 5% closeness and 90% probability criteria If the credibility standard used is the one based on the expected number of observations of [ needed, then the partial readability premium is If the credibility standard used is the one based on the expected sum of the observed values of [ needed, then the partial readability premium is Using this information, find the mean and variance of [

6 CREDIBILITY PROBLEM SET 1 SOLUTIONS 1 Since IÒ\Ó and 5 Z+<Ò\Ó, are unchanged, the standard for full credibility ill be C unchanged if 8! Ð 5 Ñ is unchanged With 5 ß T Þ* e have CÞ* Þ'%&, and C Þ'%& Þ*& 5 $Þ* With : Þ*&, CÞ*& is the Þ*(& percentile of the standard normal distribution, so that CÞ*& Þ*' In order for 8! to remain unchanged, e must have Þ*' 5 $Þ* p 5 *' Anser: D 2 For the compound Poisson distribution ith Poisson parameter (frequency distribution or number of claims per period) and claim amount distribution ] (severity distribution or amount per claim), the standard for full credibility for expected number of claims is 5 8! Ò ÐIÒ] ÓÑ ] 5] Ó 8! Ò Ð Ñ Ó Thus, ith Þ' ß ] ] 5 8 Ò Ð Ñ Ó Þ$)&)8 &!! With the coefficient of variation of ] changed to! ]! ] 5] 5]! ]! ] Þ5200, e have 8 Ò Ð Ñ Ó Þ(!%8 $(& Anser: D 3 The partial credibility factor ith 8!! is ^ Þ%! É!! 8, here 8J is the full J credibility standard for number of exposures needed The credibility factor ith 8!! 5 is ^ Þ&! É!!5 Then, É!!5 Þ&!! Þ% Þ& p 5 &'Þ& p &( Anser: C 8 J 4 The severity distribution has mean IÒ] Ó ' ' & C &C C % ß and IÒ] Ó ' ' & & & & C &C C $ ß so that Z +<Ò] Ó % Ð $ Ñ %) Þ Þ'%& With < and : Þ*, e have 8! Ð Ñ!)Þ% Þ The full credibility standard for 8! number of exposures needed for the compound Poisson distribution is Ò ÐIÒ] ÓÑ Ó The full credibility standard for the number of exposures needed for the Poisson frequency 8! distribution only is 8 Since e are considering the same Poisson frequency distribution, the value of (hich is not knon) stays the same If the same value of 8 for full credibility from the aggregate compound Poisson distribution is applied to the Poisson frequency 8! 8! distribution alone, then e set Ò ÐIÒ] ÓÑ Ó and the 8! for the Poisson frequency credibility standard must change, hich is hy it has been denoted 8! 8! 8! Z +<Ò] Ó!)Þ% &Î%) Then Ò ÐIÒ] ÓÑ Ó Ò Ð&Î%Ñ Ó p 8! &%Þ&( With T Þ*&, C: Þ*', and then in order for this to be the proper 8! for T Þ*&, e must Þ*' have &%Þ&( Ð < Ñ p < (( Anser: B

7 5 For the 60 exposure case, the credibility premium is )*Þ%( )!^'! QÐ ^'! Ñ, and for the 80 exposure case, *!Þ)) )&^)! QÐ ^)! Ñ We ish to find )(Þ&^!! QÐ ^!! Ñ In going from 60 to 80 exposures, the credibility factor changes from '! )! ^'! to ^)! (here ] is the severity distribution) 8 8 Ë! Ë! Œ Œ ÐIÒ] ÓÑ ÐIÒ] ÓÑ ^)! Thus, ^ É )! '! Þ&%(, and the to credibility premium equations become '! )*Þ%( )!^'! QÐ ^'! Ñ, *!Þ)) $Þ'^6! QÐ Þ&%(^'! Ñ )*Þ%()!^'! ^'! Juggling these equations results in *!Þ))$Þ'^ '! Þ&%(^ ß '! hich results in the quadratic equation &Þ((%^'! &Þ(*!^'! Þ%! p ^'! Þ&) ß Þ%'( Using ^'! Þ&) ß and substituting into the equations above, e get Q!!, and using ^'! Þ%'(, e get Q *(Þ' With ^'! Þ&), e get ^!! ^ '! É!! '! Þ'), and the ne credibility premium is ÐÞ')ÑÐ)(Þ&Ñ Ð Þ')ÑÐ!!Ñ *Þ& Þ With ^'! Þ%'(, e get ^!! ^ '! É!! '! Þ'!$, and the ne credibility premium is ÐÞ'!$ÑÐ)(Þ&Ñ Ð Þ'!$ÑÐ*(Þ'Ñ *Þ& Þ Anser: A 6 If the severity distribution has variance Z +<Ò] Ó!, then '(! 8! ÐIÒ] ÓÑ 8! If Z +<Ò] Ó ß &!!ß!!! ß IÒ] Ó ß!!! ß then the standard for full credibility of aggregate ß&!!ß!!! claims based on number of claims is 8! ÐIÒ] ÓÑ '(! Ð!!!Ñ ''(& Þ Anser: B 7 If the claim number distribution is Poisson, the full credibility standard for aggregate claim costs based on number of claims is 8! Ò ÐIÒ] ÓÑ Ó, here ] is the claim size distribution, and Þ96 8! Ð Ñ &$'Þ'%Þ We are given Z +<Ò] Ó! For the compound Poisson aggregate claims distribution, Z +<ÒWÓ IÒ] Ó p &!!!IÒ] Ó p IÒ] Ó &! p ÐIÒ] ÓÑ IÒ] Ó Z +<Ò] Ó &!! %!! p 8! Ò ÐIÒ] ÓÑ Ó Ð&$'Þ'%ÑÐ %! Ñ * Þ

8 8 When the claim number distribution is Poisson, the standard for full credibility for IÒ] Ó aggregate claims per period based on number of claims is 8! Ò ÐIÒ] ÓÑ Ó 8! Ò ÐIÒ] ÓÑ Ó, here ] is the claim amount random variable For the lognormal, IÒ] Ó /B:Ð 5 Ñ and IÒ] Ó /B:Ð 5 Ñ /B:Ð 5 Ñ 5 Therefore, ''!! 8 8 / %!! 8!/ p 8!!Þ* Ò/B:Ð 5 ÑÓ C T Since 8! Ð Þ Ñ, e have C Þ! But C is the percentile of the standard normal distribution From the normal table, e have T Ò[ Þ!Ó Þ)'%$ T, here [ µ RÐ!ß Ñ Therefore, T Þ($ 9Þ!)Þ% ÒÓ!)Þ% p ÒÓ Þ ß!)Þ%!ß )% p! Þ Then, Ò Ó ÒÓ!ÎÞ!!, and then Z +<ÐWÑ!!!!!)Þ% ÒÓ IÐRÑ &%Þ p IÐRÑ & Þ Since IÐRÑ IÐ] Ñ, e have!! &IÐ] Ñ p IÐ] Ñ! If R is Poisson, then Z+<ÐRÑ IÐRÑ & Then Z +<ÐWÑ IÐRÑ IÐ] Ñ p!!! & IÐ] Ñ p IÐ] Ñ!! p Z +<Ð] Ñ IÐ] Ñ ÒIÐ] ÑÓ!!!!!, hich is not possible If ] has an exponential distribution, then Z +<Ð] Ñ ÒIÐ] ÑÓ %!! Þ Then Z +<ÐWÑ IÐRÑ Z +<Ð] Ñ Z +<ÐRÑ ÒIÐ] ÑÓ p!!! & %!! Z +<ÐRÑ %!! p Z +<ÐRÑ Þ&, hich is not possible 10 Since R is Poisson, the full credibility standard for estimating the mean of R is either Z+<ÐRÑ!)Þ% (i)!)þ% ÒIÐR ÑÓ!)Þ% as the expected number of exposures of R (eeks) needed, or Z+<ÐRÑ (ii)!)þ% IÐRÑ!)Þ%!)Þ% as the total expected number of claims needed Since e do not kno the value of, the only standard e can apply is (ii) %!! With 400 claims in 20 eeks, the average number of claims per eek (sample mean) is R!! Using credibility standard (ii) above, the partial credibility factor is ^ É %!!!)Þ% Þ'!(*, and the partial credibility premium is ^ R Ð ^Ñ Q Þ' Þ$*Q T, here Q is the manual premium With 500 claims in 30 eeks, the average number of claims per eek (sample mean) is &!! R $! 'Þ'''( Using credibility standard (ii) above, the partial credibility factor is ^ É &!!!)Þ% Þ'(*(, and the partial credibility premium is ^ R Ð ^Ñ Q Þ$$ Þ$!$Q T Þ*, here Q is the manual premium

9 10 continued From the to equations, Þ' Þ$*Q T and Þ$$ Þ$!$Q T Þ*, e get Q &Þ!% and T )Þ!% Then, ith 550 claims in 35 eeks, e have &&! R $& &Þ(%$ Using credibility standard (ii) above, the partial credibility factor is ^ É &&!!)Þ% Þ(), and the partial credibility premium is ^ R Ð ^Ñ Q Þ! ÐÞ)(ÑÐ&Þ!%Ñ &Þ& 11 (E) is correct This is the formula at the bottom of page 514 in the MahlerDean Credibility study note Anser: E 12 When considering the compound Poisson aggregate claims distribution W ith claim size distribution ], e have three ays of setting the standard for full credibility: (i) the number of exposures (periods of W ) is 8! ÐIÒ] ÓÑ (ii) the number of claims is 8! ÐIÒ] ÓÑ, or (iii) the aggregate amount of claims is 8! IÒ]Ó IÒ] Ó We do not kno or IÒ] Ó in this case, and therefore (i) and (iii) cannot be used We are given È that ] has coefficient of variation $, so that IÒ] Ó $, and ÐIÒ] ÓÑ *, and e can use C: Þ*' standard (ii) The credibility criterion has 8! Ð < Ñ Ð Ñ &$'Þ'%, since : Þ*& and < The standard for full credibility is &$'Þ'%Ð *Ñ &ß $''Þ% as the number of claims needed Since only 1000 claims occurred in the first year, e have not met the standard for full credibility, and e apply the method of partial credibility The credibility factor ^ is ^ 738ÖÉ!!! &ß$''Þ% ß Þ&& We are trying to determine the credibility premium for aggregate claims for the second year We have only one exposure for aggregate claims, that being the first year, so W 'Þ(& million The manual premium is given to be Q& million The credibility premium for the second year is ^ W Ð ^Ñ Q ÐÞ&&ÑÐ'ß (&!ß!!!Ñ ÐÞ(%&ÑÐ&ß!!!ß!!!Ñ &ß %%'ß &! Anser: A Þ'%& 5 M 13 Class I:!)Þ% Ð Ñ Ð!!Ñ p 5 M!ß!!! IÒ\ M Ó Ð!!Ñ p IÒ\ M Ó!ß!!! Þ'%& 5 MM Class II:!)Þ% Ð Ñ Ð!!Ñ p 5 MM %!ß!!! IÒ\ MM Ó Ð!!Ñ p IÒ\ MMÓ )!ß!!! For the combined portfolio of policies, the loss per insured is \, is a mixture of \ M and \ MM, ith and eighting applied, respectively Then %!! $ $ IÒ\Ó Ð$ ÑÐ!!Ñ Ð$ ÑÐ!!Ñ $ and IÒ\ Ó Ð$ ÑÐ!ß!!!Ñ Ð$ ÑÐ)!ß!!!Ñ %!ß!!!, so that %!!!!ß!!! Z +<Ò\Ó %!ß!!! Ð $ Ñ * Þ'%& Z +<Ò\Ó Þ'%&!!ß!!!Î* The full credibility standard for \ is Ð Ñ ÐIÒ\ÓÑ Ð Ñ Ð%!!Î$Ñ $&$ Anser: E

10 14 The partial credibility approach sets the credibility factor to be Z+<Ò\Ó Þ'%& ^ 738š Ê 8 Š 8! ÐIÒ\ÓÑ ß In this example 8! Ð Ñ!)Þ% Based on the original assumption of \ being uniform on Ò!ß) Ó, and 8 &!, ) Î e get ^ Ê &! Š!)Þ% Ð) ÎÑ Þ$($ Based on the revised assumption of \ being uniform on Ò!ß Þ) Ó, and 8 (&, e get Þ%%) Î ^ Ê (& Š!)Þ% ÐÞ ) ÎÑ Þ%&'! ^ Î^ È(&Î&! È Þ& Anser: D 15 The full credibility standard ithin 5% of expected number claims per year 90% of the time has < (5%) and C: Þ'%& (95th percentile of the standard normal distribution) For the annual claim number distribution being Poisson, the number of claims (not exposures) needed for C: Þ'%& full credibility is 8! Ð < Ñ Ð Ñ!)Þ% actual number of observations The partial credibility factor (hen less than 1) is ^É number needed for full credibility We are told that ^Þ'), from hich e get actual number of observations number needed for full credibility ÐÞ')Ñ Þ%', so that actual number of observations ÐÞ%'ÑÐ!)Þ%Ñ &!!Þ& 5005 is the full credibility standard C: C: ith < Þ and C :, here &!!Þ& Ð Ñ Ð Ñ!!C :, so that C: È < Þ &Þ!!& Þ% From reference to the standard normal table, 2 $ is the 9874th percentile Therefore, the probability that the 500 observed claims ill satisfy the full credibility standard ithin 10% of expected number of claims per year is ÐÞ*)(%Ñ Þ*(%) Anser: D 16!)Þ% ÒÓ!)Þ% p ÒÓ Þ ß!)Þ%!ß )% p! Þ Then, Ò Ó ÒÓ!ÎÞ!!, and then Z +<ÐWÑ!!!!!)Þ% ÒÓ IÐRÑ &%Þ p IÐRÑ & Þ Since IÐRÑ IÐ] Ñ, e have!! &IÐ] Ñ p IÐ] Ñ!

11 17 We ill denote the frequency by R and the severity ill be ] The full credibility standard for the expected number of claims needed is Z +<ÒRÓ ÐIÒ] ÓÑ IÒRÓ Z +<Ò] Ó!)Þ% IÒRÓ ÐIÒ] ÓÑ &% Since the severity is exponential, e have IÒ] Ó ) and Z +<Ò] Ó ) ÐIÒ] ÓÑ The full credibility standard for expected number of claims needed becomes Z +<ÒRÓIÒRÓ!)Þ% IÒRÓ Since both the mean and variance of R are increasing by 20%, this full credibility standard is unchanged at 5412 Z +<Ð] Ñ Î 18 Þ (a)(i)!)þ% ÒIÐ] ÑÓ!)Þ% Ò) ÎÓ $'!Þ) Þ Z +<Ð] Ñ ) Î (ii)!)þ% IÐ] Ñ!)Þ% ) Î )!Þ%) Þ IÒ] Ó ) Î$ (b)(i)!)þ% ÒÓ!)Þ% ÒIÐ] ÑÓ!)Þ% Ò) ÎÓ %%$Þ IÒ] Ó ) Î$ (ii)!)þ%!)þ% IÐ] Ñ!)Þ% ) Î (Þ') (iii)!)þ% ÒÓ IÒRÓ %%$Þ %%$Þ ) 19 The full credibility standard based on the number of observations needed is!)þ% Z+<Ò[Ó ÐIÒ[ ÓÑ and based on the sum of the observed values it is!)þ% Z+<Ò[Ó IÒ[Ó From the given information, the sample mean of the observed values is [ The credibility premium based on partial credibility using is ^[ Ð ^ÑQ, here [ )&(Þ% ß Q!!! and ^ is the partial credibility factor $!!ß!!! $&! )&(Þ% Þ Using the credibility standard based on the expected number of observations needed, ^ Ê $&!, so that )&(Þ%^!!!Ð ^Ñ ))%Þ%!, from hich e get ÐIÒ[ ÓÑ!)Þ% Z+<Ò[Ó $&! Z+<Ò[Ó ^ Þ)!*, and therefore Z+<Ò[Ó Þ'&%, so that Þ%*%!)Þ% ÐIÒ[ ÓÑ ÐIÒ[ ÓÑ Using the credibility standard based on the expected sum of the observed values needed, ^ Ê $!!ß!!!!)Þ% Z+<Ò[Ó IÒ[Ó, so that )&(Þ%^!!!Ð ^Ñ ))(Þ*, from hich e get $!!ß!!! Z+<Ò[Ó ^ Þ(*!, and therefore Z+<Ò[Ó Þ'$, so that %%&!)Þ% IÒ[Ó IÒ[Ó Z +<Ò[ ÓÎIÒ[ Ó %%& Then, IÒ[ Ó Z+<Ò[ÓÎÐIÒ[ÓÑ Þ%*% *!, and Z +<Ò[ Ó %!ß!!!