The Reinsurance Expected Loss Cost Formula

The Reinsurance Expected outrage Cost principleELCF is the exorbitance freeing bell portion (as a percentage of total lost cost). primary care provider is the basal company/subject bounty. PCPLR is the primary company permissible outlet ratio (including any loss adjustment expenses c everyplaceed as a part of loss)RCF is the rove correction factor which is the reinsurers adjustment for the directd adequacy or inadequacy of the primary localizeGiven that the coverage of this treaty is per-occurrence, we must as well weigh the manual difference rate for the clash exposure.In order to determine the reinsurers intemperanceiveness shargon the ALAE is added to all(prenominal) deed, and then outcrys from policy limits which be below the trammel gratuity will be introduced into the excess layer.The reinsure may have stimulate entropy that describe the bi-variate statistical dispersion of indemnity and ALAE, or such information push aside be obtained from ISO or si milar organization outside of United States of America. With these data the reinsurer is able to construct the increase limits tables with ALAE added to the loss instead of residing in its entirety in the basic limits coverage.A nonher more than simple alternative is to adjust the manual increased limits factors so that they to account for the addition of the ALAE to the loss. A basic way of doing this is to occasion the as impartption that the ALAE for from each one and every claim is a deterministic function of indemnity amount for the claim, which means adding exactly % to each claim value for the reaching of claim surfaces that atomic number 18 near the layer of interest.This factor is smaller than the overall ratio of ALAE to ground-up indemnity loss, as much of the total ALAE relates to small claims or claims closed in(p) with no indemnity. self-confidence when ALAE is added to loss, every claim with indemnity greater than $300,000 = (1+ ) enters the layer $1,400,000 e xcess of $600,000, and that the loss amount in the layer reaches $1,400,000 when the ground-up indemnity reaches $2,000,000 = (1+ ).From this the quantity increased limits factors shag be modified to account forALAE added to the loss. In this liability context, Formula for RELC can be used with PCP as the basic limit premium and PCPLR can be used as the primary company permissible basic limits loss ratio. premiss Given the clash exposure an overall loss loading of % is sufficient enough to adjust the loss cost for this layer predicted from the stand-alone policies.Then ELCF determines the excess loss in the layer $1,400,000 with excess of $600,000 which arises from each policy limit and plus its contribution to the clash losses as a percentage of the basic limits loss that arise from the same policy limit.The prescript for ELCF which is evaluated at limit (Lim) is as followsFormula Liability ELCF for ALAE Added to Indemnity LossELCF(Lim) = 0WhereAttachment Point AP = $600,000Rei nsurance Limit RLim = $1,400,000 clang loading = 5% wastefulness ALAE loading = 20%The table 2 displays this method for a part of Allstates exposure using the hypothetical increased limits factors to calculate the excess loss cost factors with both ALAE and risk load excluded. instrument panel 2 b are(a) Loss Cost Factors with ALAE Added to Indemnity Loss at 20% add-on and a Clash Loading of 5%Table Excess Loss Cost Factors with ALAE Added to Indemnity Loss at 20% add-on and a Clash Loading of 5%(1)Policy Limit in $(2)ILF w/o risk load and w/o ALAE(3)ELCF200,0001.00000500,0001.24860600,0001.29420.05751,000,0001.40940.20261,666,6661.52730.35122,000,000 or more1.56870.4033Source own calculation establish on Patrik (2001)Using the Formula 4., the ELCF($600,000) = 1.20*1.05*(1.2942-1.2486) = 0.0575, and ELCF($2,000,000) =1.20*1.05*(1.5687-1.2486) = 0.4033. premise1 for this exposure the Allstates permissible basic limit loss ratio is PCPLR = 70%.Assumption2 reinsurers evaluation ind icates that the cedants rates and offsets are sufficient and therefore RCF is 1.00.The reinsurer can now calculate the exposure rate RELC and the reinsurers undiscounted estimate of loss cost in the excess layer as can be seen in the table 3.Table 3 Reinsurance Expected Loss Cost (undiscounted)Table Reinsurance Expected Loss Cost (undiscounted)(1)Policy Limitin $(2)Estimated Subject Premium Year 2009 in $(3)ManualILF(4)Estimated Basic Limit Loss Cost0.70x(2)/(3)(5)ELCF(6)RELC in $(4)x(5)Below 600,0002,000,0001.10 (avg.)1272727.2700600,0002,000,0001.351,037,037.040.057559,629.631,000,0002,000,0001.50933,333.330.2026189,093.332,000,000 or more4,000,0001.75 (avg.)1,600,000.000.3512562,920.00Total10,000,000n.a.4,843,197.64n.a.811,642.96Source own calculation based on Patrik (2001)An exposure loss cost can be estimated using probability models of the claim size distributions.This directly gives the reinsurer the claim count and the claim severity information which the reinsurer can use in the simple risk theoretic model for the join loss.Assumption the indemnity loss distribution underlying Table 2 is Pareto with q =1.1 and b =5,000. Then the simple model of adding the 20% ALAE to the indemnity per-occurrence changes the indemnity of a Pareto distribution to a new Pareto with q =1.1and b=5,000*1.20 = 6,000.The reinsurer has to adjust the layer severity for a clash and this can be through by multiplying with 1+ =1.05. The reinsurer can therefore calculate from each policy limit the excess judge claim sizes, after dividing the pass judgment claim size by the RELC for each limit the reinsurer obtains the estimates of expected claim count. This is done in Table 4.The expected claim size can be calculated as followsFirstly the expected excess claim severity over the attachment point d and subject to the reinsurance limit RLim for a policy limit can has to be calculated. This can be done as followsFor = 600,000For =1,000,000For =2,000,000The reinsurer is now able to calculate the expected claim count, the estimation can be seen in the table 4Table 4 Excess Expected Loss, Claim Severity and Claim CountTable Excess Expected Loss, Claim Severity and Claim CountPolicy Countin $(2)RELC in $(3)Expected ClaimSize in $(4)Expected Claim Count(2)/(3)600,00059,629.63113,9280.5231,000,000189,093.33423,1640.4472,000,000 or more562,920.00819,5570.687Total811,642.961,356,6491.68Source own calculation based on Patrik (2001)The total excess expected claim size for this exposure is $1,356,649.If the independence of claim events across all of the exposures can be assumed, the reinsurer can also obtain total estimates of the overall excess expected occurrence (claim) size and the expected occurrence (claim) count.Now we are going to estimate the experience rating. mistreat 3 Gather and reconcile primary claims data segregated by major rating class groups.As in the Example of property quota take treaties, the reinsurer needs the claims data separated as the expo sure data, and the reinsurer also necessitys some history of the individual large claims. The reinsurer usually receives information on all claims which are greater than one-half of the proposed attachment point, but it is important to receive as much data as possible.Assumption a claims re ensure has been performed and the reinsurer received a detailed history for each known claim larger than $100,000 occurring 2000-2010, which were evaluated 12/31/00, 12/31/01, 12/31/09, and 6/30/10. amount 4 Filter the major catastrophic claims out of the claims data.The reinsurer wants to identify clash claims and the mass tort claims which are significant. By separating out the clash claims, the reinsurer can estimate their size and their frequency and how they relate to the non-clash claims. These statistics should be compared to the values that the reinsurer knows from other cedants and therefore is able to get a better approximation for the loading.Step 5 dash the claims data to the ratin g period.As with the employment for the property-quota share treaties, the turn outing should be for the inflation and also for other changes in the exposure (e.g. higher policy limits) which may carry on the loss say-so, but unlike with the proportional coverage, this step cannot be skipped. The reason for this is the leveraged effect which has the inflation upon the excess claims. The constant inflation rate increases the aggregate loss beyond any attachment point and it increases faster than the aggregate loss below, as the claims grow into the excess layer, whereas their value below is stopped at the attachment point. Each ground-up claim value is trended at each evaluation, including ALAE, from year of occurrence to 2011. For example, consider the treatment of a 2003 claim in the table 5.Table 5 Trending an Accident Year 2003 ClaimTable Trending an Accident Year 2003 Claim(1)Evaluation Date(2)Value at Evaluation In $(3)Trend factor(4)2011 Level Value in 4(5)Excess Amount i n$12/31/0301.620012/31/0401.620012/31/05250,0001.62405,000012/31/06250,0001.62405,000012/31/07300,0001.62486,000012/31/08400,0001.62648,00048,00012/31/09400,0001.62648,00048,00006/30/10400,0001.62648,00048,000Source own calculation based on Patrik (2001)The reasoning for a single trend factor in this example is that the trend affects the claim values according to the accident date and not by an evaluation date.The trending of the policy limits is a delicate issue, because if a 2003 claim on a policy which has limit that is less than $500,000 inflates to above $600,000 ( plus ALAE), will be the policy limit that will be sold in the year 2011 greater than $500,000?It seems that over long periods of time, that the policy limits change with inflation.Therefore the reinsurer should over time, if possible, receive information on the Allstates policy limit distributions.Step 6 Develop the claims data to settlement values.The next step is to construct the historical accident year, thus we w ant to develop the year triangles for each fictitious character of a large claim from the data which was produced in column (5) of Table 5. Typically all claims should be combined together by major line of melody. Afterwards the loss knowledge factors should be estimated and applied on the excess claims data while using the standard methods. Also in order to check for reasonableness and comparable coverages we want to compare the increase patterns that were estimated from Allstates data to our own expectations which have their basis in our own historical data. When considering the claim in Table 5 we see that sole(prenominal) $48,000 is over the attachment point, and also only at the fifth festering pointTable 6 Trended Historical Claims in the Layer $1,400,000 Excess of $600,000 (in $1,000s)Table Trended Historical Claims in the Layer $1,400,000 Excess of $600,000 (in $1,000s)Assumption our triangle looks like the Table 6Acc. YearAge 1 in $Age 2 in $Age 3 in $Age 9 in $Age 1 0 in $Age 10.5 in $200009026425935135120010015476379820087711725620090020100ATA4.3361.5731.1661,349n.a.n.a.ATU15.0363.5472.3451.4011.050= tailSmoothed Lags11.9%28.7%47.7%93.1%95.3%96.7%Source own calculation based on Patrik (2001)WhereATA is Age-To-Age development factorATU is Age-To-Ultimate development factorLag(t) is the percentage of loss reported at time tThe selection of the tail factor of 1.05 is based upon the general information about the development for this type of an exposure beyond ten years.By changing to the inverse for the point of view from the age-to- supreme factors, the time drop offs of the claim dollar reporting, the loss reporting view is transformed to that of the cumulative distribution function (CDF) whose domain is 0,), this transformation gives a better outlook of the loss development pattern. It also allows considering and measuring the average (expected) lag and some other moments, that are comparable to the moments of loss development patterns from o ther exposures.Given the chaotic development of excess claims, it is a important to employ smoothing technique. If the smoothened factors are correctly estimated they should more credible loss development estimates which are more credible. They also allow to evaluate the function Lag( ) at every supreme time.The smoothing which was introduced in the last row of Table 6 is based on a Gamma distribution with a mean of 4 (years) and a standard deviation of 3.It is also usually useful to analyze the large claim paid data, if possible, both to estimate the patterns of the excess claims earnings and also to supplement the ultimate estimates which are based only on the reported claims that were used above.Sometimes the only data available are the data on aggregate excess claims, which would be the historical accident year per development year $1,400,000 excess of $600,000 aggregate loss triangle. Pricing without specific information about the large claims in such a situation, is very ris ky, but it is occasionally done.Step 7 Estimate the catastrophic loss potential.The mass tort claims such as pollution clean-up claims distort the historical data and therefore need special treatment. As with the property coverage, the analysis ofAllstates exposures may allow us to predict some capable loading for the future mass tort claim potential.As was said in the Step 4, the reinsurer needs to identify the clash claims.With the separation of the clash claims, for each claim, the various parts are then added together to be applied to the occurrence loss amount at the attachment point and at the reinsurance limit. If it is not possible to identify the clash claims, then the estimation of the experience of RELC has to include a clash loading which is based on judgment of the general type of exposure.Step 8 Adjust the historical exposures to the rating period.As in the example on the property quota-share treaties the historical exposure (premium) data has to be change in such a manner that makes the data are reasonably relevant to the rating period, therefore the trending should be for the primary rate, for the underwriting changes and also for other changes in exposure that may affect the loss potential of the treaty..Step 9 Estimate an experience expected loss cost, PVRELC, and, if desirable, a loss cost rate, PVRELC/PCP.Assumption we have trended and developed excess losses for all classes of Allstates casualty exposure. The standard practice is to add the pieces up as seen in the table 7.Table 7 Allstate insurance Company Casualty BusinessTable Allstate Insurance Company Casualty Business(1)Accident Year(2)Onlevel PCP in $(3)Trended and Developed Loss and Excess Loss (estimated RELC) in $(4)Estimated Cost Rate in %(3)/(2)2002171,6946,7143.912003175,9069,2885.282004178,15213,5227.592005185.89410,8205.822006188,3449,1344.582007191,3486,6583.4820081971228,5364.332009198,45212,8406.47201099,5002,8262.84Total1,586,41280,3365.06Total w/o 20101,486,91277,5 105.21Source own calculation based on Patrik (2001)The average loss cost rate for eight years is 5.21%, where the data from the year 2010 was eliminated as it is too green (undeveloped) and there does not seem to be a particular trend from year to year.Table 7 gives us the experience-based estimate, RELC=PCP =5.21%, but this estimate has to be loaded for the existing mass tort exposure, and also for the clash claims if we had insufficient information on the clash claims in the claims data.Step 10 Estimate a credibility loss cost or loss cost rate from the exposure and experience loss costs or loss cost ratesThe experience loss cost rate has to be weighed against the exposure loss cost rate that we already calculated. If there is more than one answer with different various answers that cannot be further reconciled, the last answers for the $1.400, 000 excess of $600,000 claim count and for the severity may be based on the credibility balancing of these separate estimates. All the d ifferences should however not be ignored, but should be included in the estimates of the parameter (and model) uncertainty, and therefore providing a rise to a more realistic measures of the variances, etc., and of the risk.Assumption simple situation, where there are weighed together only the experience loss cost estimate and the exposure loss cost estimate. The hexad considerations for deciding on how much weight should be given to the exposure loss cost estimate areThe accuracy of the estimate of RCF, the primary rate correction factor, and thus the accuracy of the primary expected loss cost or loss ratioThe accuracy of the predicted distribution of subject premium by line of businessFor excess coverage, the accuracy of the predicted distribution of subject premium by increased limits table for liability, by state for workers compensation, or by type of insured for property, within a line of businessFor excess coverage, the accuracy of the predicted distribution of subject premi um by policy limit within increased limits table for liability, by hazard group for workers compensation, by amount insured for propertyFor excess coverage, the accuracy of the excess loss cost factors for coverage above the attachment pointFor excess coverage, the degree of potential exposure not contemplated by the excess loss cost factorsThe credibility of the exposure loss cost estimation decreases if there are problems with any of these six items listed.Also the six considerations from which can be decided how much weight can be given to the experience loss cost estimate areThe accuracy of the estimates of claims cost inflationThe accuracy of the estimates of loss developmentThe accuracy of the subject premium on-level factorsThe stability of the loss cost, or loss cost rate, over timeThe possibility of changes in the underlying exposure over timeFor excess coverage, the possibility of changes in the distribution of policy limits over timeThe credibility of the experience loss cost estimate lessens with problems with any of the six items.Assumption the credibility loss cost rate is RELC/PCP = 5.75%.For each of the exposure sept a loss discount factor is estimated, which is based on the expected loss payment pattern for the exposure in the layer $1,400,000 excess of $600,000, and on a chosen investment yield. Most actuaries support the use of a risk-free yield, such as U.S. Treasuries for U.S. business, for the approximation of the maturity of the average claim payment lag. Discounting is significant only for longer tail business.On a practical base for a bond maturity which is between five to ten years it is better to use a single, constant fixed rate.Assumption the overall discount factor for the loss cost rate of 5.75% is RDF= 75%, which gives PVRELC/PCP = RDF*RELC/PCP =0.75*5.75%= 4.31%, or PVRELC= 4.31% * $200,000,000 = $8,620,000.The steps 11 and 12 with this example are reversed.Step 12 Specify values for RCR, RIXL, and RTERAssumption the standard guidelines for this size and type of a contract and this type of an exposure specify RIXL = 5% and RTER = 15%.The reinsurance pure premium RPP can be calculated as RPP = PVRLC/(1-RTER) = $8,620,000/0.85 = $10,141,176 with an expected profit as RPP PVRELC = $10,141,176 $8,620,000 = $1,521,176 for the risk transfer. As the RCR = 0% we can calculate the technical reinsurance premium of RP = RPP/(1-RIXL) = $10,141,176 /0.95 = $10,674,922. This technical premium is therefore above the maximum of $10,000,000 which was specified by the Allstate Insurance Company.If there is nothing wrong with technical calculations, then the reinsurer has deuce options. The first one is to accept the expected reinsurance premium of $10,000,000 at a rate of 5%, with the expected profit reduced to $10,000,000 $8,620,000 = $1,380,000Or secondly the reinsurer can propose a variable rate contract, with the reinsurance rate varying due to the reinsurance loss experience, which in this case is a retrospectivel y rated contract.As the Allstate Insurance Company is asking for a retrospectively rated contract we select the second possibility. To construct a plum and balanced rating plan, the distribution of the reinsurance of an aggregate loss has to be estimated. Now we proceed with step 11.Step 11 Estimate the probability distribution of the aggregate reinsurance loss if desirable, and perhaps other distributions such as for claims payment timing.In this step the Gamma distribution approximation will be used. As our example is lower (excess) claim frequency situation, the standard risk theoretic model for aggregate losses will be used together with the first two moments of the claim count and the claim severity distributions to approximate the distribution of aggregate reinsurance loss.The aggregate loss in the standard model is written as the sum of the individual claims, as follows.Formula Aggregate LossL=X1 + X2 ++ XNwithL as a random variable (rv) for aggregate lossN as a rv for numb er of claims (events, occurrences)Xi as rv for the dollar size of the ith claimThe N and Xi are referring to the amount of the ith claim and to the excess number of claims.To see how the standard risk theoretic model relates to the distributions of L, N and the Xis see Patrik (2001). We are working with the assumption that the Xis are both identically and one by one distributed and also independent of N, further we assume that the kth moment of L is determined completely by the first k moments of N and the Xis. There is following relationships.Formula First Two Central Moments of the Distribution of Aggregate Loss under the Standard Risk Theoretic ModelEL = EN x EXVarL = EN x EX2 + (VarN EN) x EX2Assumption the EL = RELC =5.75%*$200,000,000 = $11,500,000 (undiscounted).We assume simplistically independent and identical distribution of the excess claim sizes and also the independency of the excess claim (occurrence) count. Usually this is a reasonable assumption.For our layer $1,4 00,000 excess of $600,000, our modeling assumptions and results are shown in the formula below.Formula Allstate $1,400,000 Excess of $600,000 Aggregate Loss Modeling Assumptions and Results